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For an integer 1, m ≥ we study cyclic codes of length ℓ over a commutative non-chain ring F2m + uF2m, where u² = u. With a new Gray map and Euclidean dual-containing cyclic codes, we provide many new and superior codes to the best-known quantum error-correcting codes. Also, we characterise LCD codes of length ℓ with respect to their generator polynomials and prove that F2m--image of an LCD code of length ℓ is an LCD code of length 2ℓ. Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m. [ABSTRACT FROM AUTHOR]

In this study, we define the structure formation of the annihilator monic prime graph of commutative rings, whose distinct vertices X and J satisfies a condition a n n X J ≠ a n n X ⋃ a n n (J) , graph is denoted by AMPG (Z n [ x ] / 〈 f x 〉). The chromatic number of the annihilator monic prime graph is determined to establish the color (signless laplacian) energy of the graph. Also, we introduce color based energy of the annihilator monic prime graph, namely color distance signless Laplacian energy. Diag (T r) + D (G) + A χ , where Diag (T r) denotes the diagonal matrix of the vertex transmissions in the graph, D (G) is the distance matrix of vertices whose diagonals are zero, the entries of the matrix A χ are 1 if η i and η j are adjacent, −1 if η i and η j are non-adjacent with different colors, otherwise 0. The eigenvalues of color distance signless Laplacian matrix will be denoted by ∂ 1 χ + ≤ ∂ 2 χ + ≤ ∂ 3 χ + ≤ ⋯ ≤ ∂ n χ + , then DS L χ + of annihilator monic prime graph are define as E D S L χ + Z n x 〈 f x 〉 = ∑ 1 n ′ ∂ i χ + - t G where t (G) is the transmission of the graph. Based on the eigenvalues, we can study some more properties of graphs. Applications of color based energy are also discuss in this paper. [ABSTRACT FROM AUTHOR]

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In [1], Anscombe and Koenigsmann give an existential ∅-definition of the ring of formal power series F[[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability results for henselian valued fields with finite or pseudo-algebraically closed residue fields. [ABSTRACT FROM PUBLISHER]

We extend the construction of bad fields of characteristic zero to the case of arbitrary prescribed divisible green torsion. [ABSTRACT FROM PUBLISHER]

We characterize polynomials / with integer coefficients such that a ring with unity R is necessarily commutative if f(x) is central for all x ∉ R. We also solve the corresponding problem without the assumption that the ring has a unity. [ABSTRACT FROM AUTHOR]

Abstract: Inversive meadows are commutative rings with a multiplicative identity element and a total multiplicative inverse operation satisfying . Divisive meadows are inversive meadows with the multiplicative inverse operation replaced by a division operation. We give finite equational specifications of the class of all inversive meadows and the class of all divisive meadows. It depends on the angle from which they are viewed whether inversive meadows or divisive meadows must be considered more basic. We show that inversive and divisive meadows of rational numbers can be obtained as initial algebras of finite equational specifications. In the spirit of Peacockʼs arithmetical algebra, we study variants of inversive and divisive meadows without an additive identity element and/or an additive inverse operation. We propose simple constructions of variants of inversive and divisive meadows with a partial multiplicative inverse or division operation from inversive and divisive meadows. Divisive meadows are more basic if these variants are considered as well. We give a simple account of how mathematicians deal with 1/0, in which meadows and a customary convention among mathematicians play prominent parts, and we make plausible that a convincing account, starting from the popular computer science viewpoint that 1/0 is undefined, by means of some logic of partial functions is not attainable. [Copyright &y& Elsevier]

A meadow is a commutative ring with an inverse operator satisfying 0-1 = 0. We determine the initial algebra of the meadows of characteristic 0 and prove a normal form theorem for it. As an immediate consequence we obtain the decidability of the closed term problem for meadows and the computability of their initial object. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notions of weakly Γ-ring and Γ-field and give some characterization of Γfields. We also define the maximal ideal in a weakly Γ-ring and prove that the ideal of a commutative weakly Γ-ring M is maximal if and only if the quotient weakly Γ-ring M/I is a Γ-field. [ABSTRACT FROM AUTHOR]