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Let A ⊆ B be commutative rings with unity, let S ⊆ T be torsion-free cancellative monoids, and let n ≥ 1 . We give a characterization of when the monoid ring A [ S ] is n-root closed in the monoid ring B [ T ] . For torsion-free cancellative ordered monoids (S , ≤) ⊆ (T , ≤) , we also present sufficient conditions and necessary conditions for the generalized power series ring [ ​ [ A S , ≤ ] ​ ] to be n-root closed in the generalized power series ring [ ​ [ B T , ≤ ] ​ ] . [ABSTRACT FROM AUTHOR]

A ring R is called right cP -Baer if the right annihilator of a cyclic projective right R-module in R is generated by an idempotent. These rings are a generalization of the right p.q.-Baer rings and abelian rings. Following Birkenmeier and Heider (Commun Algebra 47(3):1348–1375, 2019 ), we investigate the transfer of the cP -Baer property between a ring R and many polynomial extensions (including skew polynomials, skew Laurent polynomials, skew power series, skew inverse Laurent series), and monoid rings. As a consequence, we answer a question posed by Birkenmeier and Heider (2019). [ABSTRACT FROM AUTHOR]

Let R 1 ⊆ R 2 be commutative rings with unity and let S 1 ⊆ S 2 be torsion-free abelian monoids. We give a characterization of when the extension R 1 [ S 1 ] ⊆ R 2 [ S 2 ] of monoid rings is a root extension. For torsion-free, ≤ -cancellative, strictly ordered abelian monoids (S 1 , ≤) ⊆ (S 2 , ≤) , we also give a characterization of when the extension [ ​ [ R 1 S 1 , ≤ ] ​ ] ⊆ [ ​ [ R 2 S 2 , ≤ ] ​ ] of generalized power series rings is a root extension. [ABSTRACT FROM AUTHOR]

The w-operation, which is in some respects a "better behaved" variant of the classic t-operation, has recently been an object of intense study. In this article, we introduce and study general w-ZPI-rings, which are commutative rings where every proper w-ideal is a w-product of prime w-ideals. We give several characterizations of general w-ZPI-rings and investigate when a monoid ring R [ S ] with S cancellative is a general w-ZPI-ring. On the way to answering the latter question, we formulate a reusable tool for reducing certain monoid ring classification problems to the monoid domain special case. [ABSTRACT FROM AUTHOR]

The decoding algorithm for the correction of errors of arbitrary Mannheim weight has discussed for Lattice constellations and codes from quadratic number fields. Following these lines, the decoding algorithms for the correction of errors of n = p-1 2 length cyclic codes (C) over quaternion integers of Quaternion Mannheim (QM) weight one up to two coordinates have considered. In continuation, the case of cyclic codes of lengths n = p-1 2 and 2n - 1 = p - 2 has studied to improve the error correction efficiency. In this study, we present the decoding of cyclic codes of length n = ϕ (p) = p - 1 and length 2n - 1 = 2ϕ (p) - 1 = 2p - 3 (where p is prime integer and ϕ is Euler phi function) over Hamilton Quaternion integers of Quaternion Mannheim weight for the correction of errors. Furthermore, the error correction capability and code rate tradeoff of these codes are also discussed. Thus, an increase in the length of the cyclic code is achieved along with its better code rate and an adequate error correction capability. [ABSTRACT FROM AUTHOR]

Let D be a domain and let S be a torsion-free monoid such that D has characteristic 0 or the quotient group of S satisfies the ascending chain condition on cyclic subgroups. We give a characterization of when the monoid algebra D [ S ] is weakly Krull. As corollaries, we reobtain the results on when D [ S ] is Krull resp. weakly factorial, due to Chouinard resp. Chang. Furthermore, we deduce a characterization of generalized Krull monoid algebras analogous to our main result and we characterize the weakly Krull domains among the affine monoid algebras. [ABSTRACT FROM AUTHOR]

Several different generalizations of finite factorization domains (i.e., integral domains where every nonzero nonunit has only finitely many divisors up to associates) have been defined for commutative rings with zero divisors. We study these notions in the context of commutative monoid rings with zero divisors, utilizing semigroup theory to simultaneously generalize and extend many past results about "finite factorization" properties in commutative polynomial rings. Along the way, we expand upon the general theory of factorization in commutative rings with zero divisors, providing new characterizations and results about several kinds of "finite factorization rings." [ABSTRACT FROM AUTHOR]

Several authors have studied the question of when the monoid ring DM of a monoid M over a ring D is a right and/or left fir (free ideal ring), a semifir, or a 2-fir (definitions recalled in §1). It is known that for M nontrivial, a necessary condition for any of these properties to hold is that D be a division ring. Under that assumption, necessary and sufficient conditions on M are known for DM to be a right or left fir, and various conditions on M have been proved necessary or sufficient for DM to be a 2-fir or semifir. A sufficient condition for DM to be a semifir is that M be a direct limit of monoids which are free products of free monoids and free groups. Warren Dicks has conjectured that this is also necessary. However F. Cedó has given an example of a monoid M which is not such a direct limit, but satisfies all the known necessary conditions for DM to be a semifir. It is an open question whether for this M, the rings DM are semifirs.We note here some reformulations of the known necessary conditions for a monoid ring DM to be a 2-fir or a semifir, motivate Cedó’s construction and a variant thereof, and recoverCedó’s results for both constructions. Any homomorphism from a monoid M into Z induces a Z-grading on DM, and we show that for the two monoids just mentioned, the rings DM are “homogeneous semifirs” with respect to all such nontrivial Z-gradings; i.e., have (roughly) the property that every finitely generated homogeneous one-sided ideal is free of unique rank. If M is a monoid such that DM is an n-fir, and N a “well-behaved” submonoid of M, we prove some properties of the ring DN. Using these, we show that for M a monoid such that DM.

Several different versions of "factoriality" have been defined for commutative rings with zero divisors. We apply semigroup theory to study these notions in the context of a commutative monoid ring R[S], determining necessary and sufficient conditions for R[S] to be various kinds of "unique factorization rings." Our work generalizes Anderson et al.'s results about "unique factorization" in R[X], Gilmer and Parker's characterization of factorial monoid domains, and Hardy and Shores's classification of when R[S] is a principal ideal ring (for S cancellative). Along the way, we determine when R[S] is "restricted cancellative" or satisfies various "(restricted) ideal cancellation laws." [ABSTRACT FROM AUTHOR]

In this paper, the ring of polynomials is studied in a systematic way through the theory of monoid rings. As a consequence, this study provides canonical approaches in order to find easy and rigorous proofs and methods for many facts on polynomials and formal power series; some of them as sample are treated in this paper. Besides the universal properties of the monoid rings and polynomial rings, a universal property for the formal power series rings is also established. [ABSTRACT FROM AUTHOR]

In this article, we investigate the concept of radical factorization with respect to finitary ideal systems of cancellative monoids. We present new characterizations for r-almost Dedekind r-SP-monoids and provide specific descriptions of t-almost Dedekind t-SP-monoids and w-SP-monoids. We show that a monoid is a w-SP-monoid if and only if the radical of every nontrivial principal ideal is t-invertible. We characterize when the monoid ring is a w-SP-domain and describe when the *-Nagata ring is an SP-domain for a star operation * of finite type. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity and Z*(R) be its set of non-zero zero-divisors. The undirected zero-divisor graph of R, denoted by Γ(R), is the graph whose vertices are the non-zero zero-divisors of R, and where two distinct vertices r and s are adjacent if and only if rs = 0 or sr = 0. The distance between vertices a and b is the length of the shortest path connecting them, and the diameter of the graph, diam(Γ(R)), is the superimum of these distances. In this paper, first we prove some results about Γ(R) of a semi-commutative ring R. Then, for a reversible ring R and a unique product monoid M, we prove 0 ≤  diam (Γ (R)) ≤  diam (Γ (R [ M ])) ≤ 3. We describe all the possibilities for the pair diam(Γ(R)) and diam(Γ(R[M])), strictly in terms of the properties of a ring R, where R is a reversible ring and M is a unique product monoid. Moreover, an example showing the necessity of our assumptions is provided. [ABSTRACT FROM AUTHOR]

In this work, we introduce a method by which it is established that; how a sequence of non-primitive BCH codes can be obtained by a given primitive BCH code. For this, we rush to the out of routine assembling technique of BCH codes and use the structure of monoid rings instead of polynomial rings. Accordingly, it is gotten that there is a sequence $\{C_{b^{j}n}\}_{1\leq j\leq m}$, where $b^{j}n$ is the length of $C_{b^{j}n}$, of non-primitive binary BCH codes against a given binary BCH code $C_{n}$ of length $n$. Matlab based simulated algorithms for encoding and decoding for these type of codes are introduced. Matlab provides built in routines for construction of a primitive BCH code, but impose several constraints, like degree $s$ of primitive irreducible polynomial should be less than $16$. This work focuses on non-primitive irreducible polynomials having degree $bs$, which go far more than 16.

In this paper, we present constructions of primitive and non-primitive BCH codes using monoid rings over the local ring ℤ 2 m , with m ≥ 1. We show that there exist two sequences { C b j n } j ≥ 1 and { C b j n ′ } j ≥ 1 of non-primitive BCH codes (over ℤ 2 and ℤ 2 m , respectively) against primitive BCH codes C n of length n and C n ′ (over ℤ 2 and ℤ 2 m ), respectively. A technique is developed in an innovative way that enables the data path to shift instantaneously during transmission via the coding schemes of C n , C n ′ , { C b j n } j ≥ 1 and { C b j n ′ } j ≥ 1 . The selection of the schemes is subject to the choice of better code rate or better error-correction capability of the code. Finally, we present a decoding procedure for BCH codes over Galois rings, which is also used for the decoding of BCH codes over Galois fields, based on the modified Berlekamp–Massey algorithm. [ABSTRACT FROM AUTHOR]

Abstract For a commutative Noetherian ring R of dimension d and a commutative cancellative monoid M , the elementary action on unimodular n -rows over the monoid ring R [ M ] is transitive for n ≥ max ⁡ (d + 2 , 3). The starting point is the case of polynomial rings, considered by A. Suslin in the 1970s. The main result completes a project, initiated in the early 1990s, and suggests a new direction in the study of K -theory of monoid rings. [ABSTRACT FROM AUTHOR]

It is one of the oldest research topics in computer algebra to determine the equivalence of Riemann tensor indexed polynomials. However, it remains to be a challenging problem since Gröbner basis theory is not yet powerful enough to deal with ideals that cannot be finitely generated. This paper solves the problem by extending Gröbner basis theory. First, the polynomials are described via an infinitely generated free commutative monoid ring. The authors then provide a decomposed form of the Gröbner basis of the defining syzygy set in each restricted ring. The canonical form proves to be the normal form with respect to the Gröbner basis in the fundamental restricted ring, which allows one to determine the equivalence of polynomials. Finally, in order to simplify the computation of canonical form, the authors find the minimal restricted ring. [ABSTRACT FROM AUTHOR]

Let M be a monoid. A ring R is called M- π-Armendariz if whenever α = a g + a g + · · · + a g, β = b h + b h + · · · + b h ∈ R[ M] satisfy αβ ∈ nil( R[ M]), then a b ∈ nil( R) for all i, j. A ring R is called weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical. In this paper, we consider some extensions of M- π-Armendariz rings and further investigate their properties under the condition that R is weakly 2-primal. We prove that if R is an M- π-Armendariz ring then nil( R[ M]) = nil( R)[ M]. Moreover, we study the relationship between the weak zip-property (resp., weak APP-property, nilpotent p.p.-property, weak associated prime property) of a ring R and that of the monoid ring R[ M] in case R is M- π-Armendariz. [ABSTRACT FROM AUTHOR]

An associative ring is said to be right strongly Hopfian if the chain of right annihilators $$r_R(a)\subseteq r_R(a^2)\subseteq \cdots $$ stabilizes for each $$a\in R$$ . In this article, we are interested in the class of right strongly Hopfian rings and the transfer of this property from an associative ring R to the Ore extension $$R[x;\alpha ,\delta ]$$ and the monoid ring R[ M]. It is proved that if R is $$(\alpha ,\delta )$$ -compatible and $$R[x;\alpha ,\delta ]$$ is reversible, then the Ore extension $$R[x;\alpha ,\delta ]$$ is right strongly Hopfian if and only if R is right strongly Hopfian, and it is also shown that if M is a strictly totally ordered monoid and R[ M] is a reversible ring, then the monoid ring R[ M] is right strongly Hopfian if and only if R is right strongly Hopfian. Consequently, several known results regarding strongly Hopfian rings are extended to a more general setting. [ABSTRACT FROM AUTHOR]

BCH codes are one of the most important classes of cyclic codes for error correction. In this study, we generalize BCH codes using monoid rings instead of a polynomial ring over the binary field F. We show the existence of a non-primitive binary BCH code C of length bn, corresponding to a given length n binary BCH code C. The value of b is investigated for which the existence of the non-primitive BCH code C is assured. It is noticed that the code C is embedded in the code C. Therefore, encoding and decoding of the codes C and C can be done simultaneously. The data transmitted by C can also be transmitted by C. The BCH code C has better error correction capability whereas the BCH code C has better code rate, hence both gains can be achieved at the same time. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with total quotient ring K. Each monomorphic R-module endomorphism of a cyclic R-module is an isomorphism if and only if R has Krull dimension 0. Each monomorphic R-module endomorphism of R is an isomorphism if and only if R = K. We say that R has property (*) if for each nonzero element a ∈ R, each monomorphic R-module endomorphism of R/Ra is an isomorphism. If R has property (*), then each nonzero principal prime ideal of R is a maximal ideal, but the converse is false, even for integral domains of Krull dimension 2. An integral domain R has property (*) if and only if R has no R-sequence of length 2; the "if:" assertion fails in general for non-domain rings R. Each treed domain has property (*), but the converse is false. [ABSTRACT FROM AUTHOR]

Let D be an integral domain satisfying ACCP. We refine the classical notion of (factorization) length by recursively defining the length of a nonzero element to be the least ordinal strictly greater than the lengths of its proper divisors. This gives a surjective function L : D ⁎ → L ( D ) , where L ( D ) , called the length of D , is the least ordinal strictly greater than the length of any nonzero element. We show that an ordinal is the length of a domain satisfying ACCP if and only if it is of the form ω β . We give some conditions for when monoid domains, generalized power series domains, inert extensions, or localizations at splitting sets satisfy ACCP, and calculate the lengths of these domains in these cases. Finally, for each positive integer n ≥ 2 and each ordinal μ ≥ n , we construct a domain D satisfying ACCP and an x ∈ D ⁎ with L ( x ) = μ and l ( x ) = n , where l ( x ) denotes the number of factors in a minimum length atomic factorization of x . [ABSTRACT FROM AUTHOR]

The model-theoretic Grothendieck ring of a first order structure, as defined by Krajicěk and Scanlon, captures some combinatorial properties of the definable subsets of finite powers of the structure. In this paper we compute the Grothendieck ring, K 0 ( M R ) , of a right R -module M , where R is any unital ring. As a corollary we prove a conjecture of Prest that K 0 ( M R ) is non-trivial, whenever M is non-zero. The main proof uses various techniques from simplicial homology and lattice theory to construct certain counting functions. [ABSTRACT FROM AUTHOR]

Several authors have studied the question of when the monoid ring $$DM$$ of a monoid $$M$$ over a ring $$D$$ is a right and/or left fir (free ideal ring), a semifir, or a $$2$$ -fir (definitions recalled in §1). It is known that for $$M$$ nontrivial, a necessary condition for any of these properties to hold is that $$D$$ be a division ring. Under that assumption, necessary and sufficient conditions on $$M$$ are known for $$DM$$ to be a right or left fir, and various conditions on $$M$$ have been proved necessary or sufficient for $$DM$$ to be a $$2$$ -fir or semifir. A sufficient condition for $$DM$$ to be a semifir is that $$M$$ be a direct limit of monoids which are free products of free monoids and free groups. Warren Dicks has conjectured that this is also necessary. However F. Cedó has given an example of a monoid $$M$$ which is not such a direct limit, but satisfies all the known necessary conditions for $$DM$$ to be a semifir. It is an open question whether for this $$M,$$ the rings $$DM$$ are semifirs. We note here some reformulations of the known necessary conditions for a monoid ring $$DM$$ to be a $$2$$ -fir or a semifir, motivate Cedó's construction and a variant thereof, and recover Cedó's results for both constructions. Any homomorphism from a monoid $$M$$ into $$\mathbb {Z}$$ induces a $$\mathbb {Z}$$ -grading on $$DM,$$ and we show that for the two monoids just mentioned, the rings $$DM$$ are 'homogeneous semifirs' with respect to all such nontrivial $$\mathbb {Z}$$ -gradings; i.e., have (roughly) the property that every finitely generated homogeneous one-sided ideal is free of unique rank. If $$M$$ is a monoid such that $$DM$$ is an $$n$$ -fir, and $$N$$ a 'well-behaved' submonoid of $$M,$$ we prove some properties of the ring $$DN.$$ Using these, we show that for $$M$$ a monoid such that $$DM$$ is a $$2$$ -fir, mutual commutativity is an equivalence relation on nonidentity elements of $$M,$$ and each equivalence class, together with the identity element, is a directed union of infinite cyclic groups or of infinite cyclic monoids. Several open questions are noted. [ABSTRACT FROM AUTHOR]

Given a ringRand a monoidM, we study the concept of so called nil-Armendariz ring relative to a monoid, which is a common generalization of nil-Armendariz rings and Armendariz rings relative to a monoid. It is done by considering the nil-Armendariz condition on a monoid ringR[M] instead of the polynomial ringR[x]. We prove that several properties transfer betweenRand the monoid ringR[M], in caseRis nilM-Armendariz ring. We resolve the structure of nilM-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be nilM-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the special cases. In particular, we prove that every NI-ring is nilM-Armendariz, for any unique product monoidM. We also classify which of the standard nilpotence properties on polynomial rings pass to monoid rings. We provide various examples and classify how the nilM-Armendariz rings behaves under various ring extensions. [ABSTRACT FROM AUTHOR]

Decoding algorithms for the correction of errors for cyclic codes over quaternion integers of quaternion Mannheim weight one up to two coordinates are discussed by Özen and Güzeltepe (Eur J Pure Appl Math 3(4):670–677, 2010 ). Though, Neto et al. (IEEE Trans Inf Theory 47(4):1514–1527, 2001 ) proposed decoding algorithms for the correction of errors of arbitrary Mannheim weight. In this study, we followed the procedures used by Neto et al. and suggest a decoding algorithm for an $$n$$ n length cyclic code over quaternion integers to correct errors of quaternion Mannheim weight two up to two coordinates. Furthermore, we establish that; over quaternion integers, for a given $$n$$ n length cyclic code there exist a cyclic code of length $$2n-1$$ 2 n - 1 . The decoding algorithms for the cyclic code of length $$2n-1$$ 2 n - 1 are given, which correct errors of quaternion Mannheim weight one and two. In addition, we show that the cyclic code of length $$2n-1$$ 2 n - 1 is maximum-distance separable (MDS) with respect to Hamming distance. [ABSTRACT FROM AUTHOR]

We characterize polynomials f with integer coefficients such that a ring with unity R is necessarily commutative if f( R) = 0, in the sense that f( x) = 0 for all $${x \in R}$$. Such a polynomial must be primitive, and for primitive polynomials the condition f( R) = 0 forces R to have nonzero characteristic. The task is then reduced to considering rings of prime power characteristic and the main step towards the full characterization is a characterization of polynomials f such that R is necessarily commutative if f( R) = 0 and R is a unital ring of characteristic some power of a fixed prime p. [ABSTRACT FROM AUTHOR]

Let B[X; S] be a monoid ring with any fixed finite unitary commutative ring B and is the monoid S such that b = a + 1, where a is any positive integer. In this paper we constructed cyclic codes, BCH codes, alternant codes, Goppa codes, Srivastava codes through monoid ring . For a = 1, almost all the results contained in [16] stands as a very particular case of this study. [ABSTRACT FROM AUTHOR]

In [41], Nielsen proves that all reversible rings are McCoy and gives an example of a semicommutative ring that is not right McCoy. At the same time, he also shows that semicommutative rings do have a property close to the McCoy condition. In this article we study weak McCoy rings as a common generalization of McCoy rings and weak Armendariz rings. Relations between the weak McCoy property and other standard ring theoretic properties is considered. We also study the weak skew McCoy condition, a generalization of the standard weak McCoy condition from polynomials to skew polynomial rings. We resolve the structure of weak skew McCoy rings and obtain various necessary or sufficient conditions for a ring to be weak skew McCoy, unifying and generalizing a number of known McCoy-like conditions in the special cases. Constructing various examples, we classify how the weak McCoy property behaves under various ring extensions. As a consequence we extend and unify several known results related to McCoy rings and Armendariz rings [6, 35, 38, 43, 49]. [ABSTRACT FROM PUBLISHER]

In this article, we introduce a construction called the generalized inverse power series module M[[S-1]] over a monoid ring R[S] with coefficients in an R-module M and exponents in a commutative monoid S. This construction is a generalization of the R[x]-modules which were discussed by S. Park in [12-1412, 13, 14]. The injectivity and injective precovers of the generalized inverse power series module are considered. We also show that N is a pure submodule of M if and only if N[S] is a pure submodule of the monoid module M[S]. [ABSTRACT FROM AUTHOR]

An integral domain D is weakly integrally closed if whenever x is in the quotient field of D, and J is a nonzero finitely generated ideal of D such that xJ ⊆ J2, then x is in D. We define weakly integrally closed (WIC) numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. The characteristic function of a numerical monoid M can be thought of as an infinite binary string s(M). A pattern of finitely many 0's and 1's is called forbidden if whenever s(M) contains it, then M is not weakly integrally closed. The pattern 11011 is forbidden. We show that a numerical monoid M is WIC if and only if s(M) contains no forbidden patterns. We also show that for every finite set S of forbidden patterns, there exists a numerical monoid M that is not WIC and for which s(M) contains no stretch (in a natural sense) of a pattern in S. [ABSTRACT FROM AUTHOR]

Let M be a cancellative monoid such that the monoid ring ℤ M has no zero divisors. We show that if the monoid consisting of all elements of ℤ M with strictly positive coefficients has nonzero common right multiples, then M is left amenable. [ABSTRACT FROM AUTHOR]

An example is given of an Artinian local (commutative unital) ring R and a finite group G acting on R (via ring automorphisms) such that RG is not an Artinian ring; in this example, |G| necessarily fails to be a unit of R. Also, an example is constructed of a ring R on which an infinite cyclic group G acts such that the ring extension RG ⊆ R does not satisfy the going-down property GD; in this second example, the G-action on R is necessarily not locally finite, and it can be arranged that R is a monoid ring with any desired infinite cardinality and that RG is an integral domain with any (prime or 0) characteristic. [ABSTRACT FROM AUTHOR]

Recently, Alonso and Hermiller (2003) introduced a homological finiteness condition bi - FPn (here called “weak bi-FPn”) for monoid rings, and Kobayashi and Otto (2003) introduced a different property, also called bi - FPn (we adhere to their terminology). From these and other articles we know that: bi - FPn → left and right FPn → weak bi - FPn; the first implication is not reversible in general; the second implication is reversible for group rings. We show that the second implication is reversible in general, even for arbitrary associative algebras (Theorem 1′), and we show that the first implication is reversible for group rings (Theorem 2). We also show that the all four properties are equivalent for connected graded algebras (Theorem 4). [ABSTRACT FROM AUTHOR]

It is proved that the monoids between an archimedean, irreflexive and rigid monoid M and its universal group G, obtained by adjoining inverses to M, are archimedean, irreflexive and rigid monoids. [ABSTRACT FROM AUTHOR]