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A restricted artinian ring is a commutative ring with an identity in which every proper homomorphic image is artinian. Cohen proved that a commutative ring R is restricted artinian if and only if it is noetherian and every nonzero prime ideal of R is maximal. Facchini and Nazemian called a commutative ring isoartinian if every descending chain of ideals becomes stationary up to isomorphism. We show that every proper homomorphic image of a commutative noetherian ring R is isoartinian if and only if R has one of the following forms: (a) R is a noetherian domain of Krull dimension one which is not a principal ideal domain; (b) R ≅ D 1 × ⋯ × D k × A 1 × ⋯ × A l , where each D i is a principal ideal domain and each A i is an artinian local ring (either k or l may be zero); (c) R is a noetherian ring of Krull dimension one, simple unique minimal prime ideal 픭 , and R / 픭 is a principal ideal domain. As an application of our result, we describe commutative rings whose proper homomorphic images are principal ideal rings. Some relevant examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, we first study zero divisor graphs over finite chain rings. We determine their rank, determinant, and eigenvalues using reduction graphs. Moreover, we extend the work to zero divisor graphs over finite commutative principal ideal rings using a combinatorial method, finding the number of positive eigenvalues and the number of negative eigenvalues, and finding upper and lower bounds for the largest eigenvalue. Finally, we characterize all finite commutative principal ideal rings such that their zero divisor graphs are complete and compute the Wiener index of the zero divisor graphs over finite commutative principal ideal rings. [ABSTRACT FROM AUTHOR]

Let Hm,n be the mn2-dimensional Radford Hopf algebra over an algebraically closed field of characteristic zero. We give the classification of all ideals of 8-dimensional Radford Hopf algebra H2,2 by generators. [ABSTRACT FROM AUTHOR]

By the assumption of the theorem and the definition of the direct sum of rings, among these rows there are HT ht is the smallest possible number) that distinguish the elements ( I xb i SB 1 sb ), ..., ( I xb SB k sb i ) of ring I A i I B i , where I x i = I a i SB 1 sb , ..., I a SB n sb i , I x i I A i . Among these rows, by the hypothesis of the theorem and the definition of the direct sum of rings, there are HT ht rows (the minimal possible number) that distinguish these elements. Consider a source of faults HT ht capable of acting on a Boolean function as follows. [Extracted from the article]

For any (unital commutative) principal ideal ring A, we produce a set of A-algebra isomorphism class representatives of the ramified (unital minimal) ring extensions of A. Special attention is paid to the case of a finite principal ideal ring. The first step in this process involves sharpening some of our earlier results from the case where A is a finite special principal ideal ring. One consequence of this work is the completion of the classification, up to ring isomorphism, of the commutative (unital) rings having a unique proper (unital) subring. [ABSTRACT FROM AUTHOR]

Let H m , n be the mn2-dimensional Radford Hopf algebra over an algebraically closed field of characteristic zero. In this paper, we show that any finite dimensional indecomposable H m , n -module is a cyclic module. According to the structure of any indecomposable module under the adjoint action of H m , n , we prove that the Radford Hopf algebra H m , n is a principal ideal ring. Furthermore, we give explicitly the generators of all 28 ideals of 9-dimensional Radford Hopf algebra H 1 , 3 . [ABSTRACT FROM AUTHOR]

Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. In this paper we show that any finite-dimensional indecomposable H -module is generated by one element. In particular, any indecomposable submodule of H under the adjoint action is generated by a special element of H. Using this result, we show that the Hopf algebra H is a principal ideal ring, i.e., any two-sided ideal of H is generated by one element. As an application, we give explicitly the generators of ideals, primitive ideals, maximal ideals and completely prime ideals of the Taft algebras. [ABSTRACT FROM AUTHOR]

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]

By modifying the existing definition of a compressed zero-divisor graph Γ E (K) , we define a compressed zero-divisor graph Θ (K) of a finite commutative unital ring K , where the compression is performed by means of the associatedness relation (a refinement of the relation used in the definition of Γ E (K)). We prove that this is the best possible compression which induces a functor Θ , and that this functor preserves categorial products (in both directions). We use the structure of Θ (K) to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if R is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then R is right Noetherian. Also, we show a von Neumann regular (semiregular) ring R with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring. [ABSTRACT FROM AUTHOR]

Let D be a commutative domain with identity, and let L (D) be the lattice of nonzero ideals of D. Say that D is ideal upper finite provided L (D) is upper finite, that is, every nonzero ideal of D is contained in but finitely many ideals of D. Now let κ > 2 ℵ 0 be a cardinal. We show that a domain D of cardinality κ is ideal upper finite if and only if D is a Dedekind domain. We also show (in ZFC) that this result is sharp in the sense that if κ is a cardinal such that ℵ 0 ≤ κ ≤ 2 ℵ 0 , then there is an ideal upper finite domain of cardinality κ which is not Dedekind. [ABSTRACT FROM AUTHOR]

Let Γ (R) ¯ be the complement of the zero-divisor graph of a finite commutative ring R. In this paper, we provide the answer of the question (ii) raised by Osba and Alkam in [11] and prove that Γ (R) ¯ is a divisor graph if R is a local ring. It is shown that when R is a product of two local rings, then Γ (R) ¯ is a divisor graph if one of them is an integral domain. Further, if | A s s (R) | = 2 , then Γ (R) ¯ is a divisor graph. [ABSTRACT FROM AUTHOR]

A generalization of Köthe rings is the family of rings whose ideals are direct sums of cyclic modules. These rings were previously studied in the commutative local case. This motivated us to study commutative rings with local dimension whose ideals are direct sums of cyclic modules. First, we obtain an structure theorem for rings of local dimension ω. Then we conclude that, for a ring of local dimension ω , every ideal of R is a direct sum of cyclic modules if and only if every indecomposable ideal of R is cyclic if and only if every maximal ideal of R / Soc (R) is cyclic if and only if every maximal ideal of R is cyclic if and only if R is a principal ideal ring. [ABSTRACT FROM AUTHOR]

Because of its interesting applications in coding theory, cryptography, and algebraic combinatoris, in recent decades a lot of attention has been paid to the algebraic structure of the ring of polynomials R[x], where R is a finite commutative ring with identity. Motivated by this popularity, in this paper we determine when R[x] is a principal ideal ring. In fact, we prove that R[x] is a principal ideal ring if and only if R is a finite direct product of finite fields

Let R be a finite commutative principal ideal ring with unity. In this article, we prove that the zero-divisor graph г(R) is a divisor graph if and only if R is a local ring or it is a product of two local rings with at least one of them having diameter less than 2. We also prove that г(R) is a divisor graph if and only if г(R[x]) is a divisor graph if and only if г(R[[x]]) is a divisor graph. [ABSTRACT FROM AUTHOR]

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By two results of Köthe and Cohen-Kaplansky we obtain that 'a commutative ring has the property that every -module is a direct sum of (completely) cyclic modules if and only if is an Artinian principal ideal ring' (an -module is called completely cyclic if each submodule of is cyclic). In this paper, we describe and study commutative rings whose proper ideals are direct sum of completely cyclic modules. It is shown that every proper ideal of a commutative ring is a direct sum of completely cyclic -modules if and only if is a principal ideal ring or is a local ring with maximal ideal such that there is an index set and a set of elements such that with each a simple -module and a principal ideal ring. [ABSTRACT FROM AUTHOR]

Many studies have been conducted to characterize commutative rings whose finitely generated modules are direct sums of cyclic modules (called FGC rings), however, the characterization of noncommutative FGC rings is still an open problem, even for duo rings. We study FGC rings in some special cases, it is shown that a local Noetherian ring R is FGC if and only if R is a principal ideal ring if and only if R is a uniserial ring, and if these assertions hold R is a duo ring. We characterize Noetherian duo FGC rings. In fact, it is shown that a duo ring R is a Noetherian left FGC ring if and only if R is a Noetherian right FGC ring, if and only if R is a principal ideal ring. [ABSTRACT FROM AUTHOR]

LetVbe a finite-dimensional vector space over a fieldk, and letWbe a 1-dimensionalk-vector space. Let ⟨,⟩:V × V → Wbe a symmetric bilinear form. Then ⟨,⟩ is calledanisotropicif for all nonzerov ∈ Vwe have ⟨ v,v ⟩ ≠ 0. Motivated by a problem in algebraic number theory, we give a generalization of the concept ofanisotropyto symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that a form is anisotropic if and only if certain forms on vector spaces are anisotropic. We will also discuss the concept ofquasi-anisotropyof a symmetric bilinear form, which has no vector space analogue. Finally, we will discuss theradical rootof a symmetric bilinear form, which does not have a vector space analogue either. All three concepts have applications in algebraic number theory. [ABSTRACT FROM AUTHOR]

Abstract: A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in extends to a monomial transformation of that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming weight or the homogeneous weight satisfy the Extension Property. Conversely, if a finite ring with the Hamming or homogeneous weight satisfies the Extension Property, then the ring is Frobenius. This paper addresses the question of a characterization of all bi-invariant weights on a finite ring that satisfy the Extension Property. Having solved this question in previous papers for all direct products of finite chain rings and for matrix rings, we have now arrived at a characterization of these weights for finite principal ideal rings, which form a large subclass of the finite Frobenius rings. We do not assume commutativity of the rings in question. [Copyright &y& Elsevier]

This paper investigates the structure of cyclically pure (or C-pure) projective modules. In particular, it is shown that a ringRis left Noetherian if and only if every C-pure projective leftR-module is pure projective. Also, over a left hereditary Noetherian ringR, a leftR-moduleMis C-pure projective if and only ifM = N⊕P, whereNis a direct sum of cyclic modules andPis a projective leftR-module. The relationship C-purity with purity and RD-purity are also studied. It is shown that ifRis a local duo-ring, then the C-pure projective leftR-modules and the pure projective leftR-modules coincide if and only ifRis a principal ideal ring. IfRis a left perfect duo-ring, then the C-pure projective leftR-modules and the pure projective leftR-modules coincide if and only ifRis left Köthe (i.e., every leftR-module is a direct sum of cyclic modules). Also, it is shown that for a ringR, if every C-pure projective leftR-module is RD-projective, thenRis left Noetherian, every p-injective leftR-module is injective and every p-flat rightR-module is flat. Finally, it is shown that ifRis a left p.p-ring and every C-pure projective leftR-module is RD-projective, thenRis left Noetherian hereditary. The converse is also true whenRis commutative, but it does not hold in the noncommutative case. [ABSTRACT FROM PUBLISHER]

Let R be a finite commutative ring with identity. We prove that if R is a principal ideal ring then the directed zero-divisor graph Γ( M( R)) is balanced and eulerian. We also find the smallest finite commutative ring R with identity such that Γ( M( R)) is not balanced. [ABSTRACT FROM AUTHOR]

This paper deals with a reducible sℓ(2,C) action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau conjecture: Suppose that sℓ(2,C) acts on the formal power series ring via (1.1). Then I( f) = ( ℓ i1) ⊕ ( ℓ i2) ⊕... ⊕ ( ℓ is ) modulo some one dimensional sℓ(2,C) representations where (ℓ i ) is an irreducible sℓ(2,C) representation of ℓ i dimension and { ℓ i1 ℓ i2,..., ℓ is } ⊆ { ℓ 1 , ℓ 2..., ℓ r }. Unlike classical invariant theory which deals only with irreducible action and 1-dimensional representations, we treat the reducible action and higher dimensional representations successively. [ABSTRACT FROM AUTHOR]

Abstract: Let be a commutative, local, and principal ideal ring with maximal ideal and residue class field . Suppose that every element of is square. Then the problem of classifying arbitrary symmetric matrices over by congruence naturally reduces, and is actually equivalent to, the problem of classifying invertible symmetric matrices over by congruence. [Copyright &y& Elsevier]

Abstract The Iwasava decomposition is proved for the Steinberg groups of types 2 A 2 l−1, 2 D l , 2 E 6, 3 D 4 over the field of fractions of a principal ideal ring. By using this decomposition, it is described that subgroups exist between the Steinberg groups over the rings D and K under some restrictions on the ring D. [ABSTRACT FROM AUTHOR]