Weakly $$\delta $$-n-ideals of commutative rings

If N is a Noetherian module over the commutative ring R (throughout the paper, R will denote a commutative ring with identity), then the study of N in many contexts can be reduced to the study of a finitely generated module over a commutative Noetherian ring, because N has a natural structure as a module over R/(0 : N) and the latter ring is Noetherian. For a long time it has been a source of irritation to me that I did not know of any method which would reduce the study of an Artinian module A over the commutative ring R to the study of an Artinian module over a commutative Noetherian ring. However, during the MSRI Microprogram on Commutative Algebra, my attention was drawn to a result of W. Heinzer and D. Lantz [2, Proposition 4.3]; this proposition proves that if A is a faithful Artinian module over a quasi-local ring (R, M) which is (Hausdorff) complete in the M-adic topology, then R is Noetherian. It turns out that a generalization of this result provides a missing link to complete a chain of reductions by which one can, for some purposes, reduce the study of an Artinian module over an arbitrary commutative ring R to the study of an Artinian module over a complete (Noetherian) local ring; in the latter situation we have Matlis’s duality available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.KeywordsLocal RingMaximal IdealCommutative RingHomomorphic ImageNoetherian RingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Torsion theories over commutative rings

A commutative ring R with identity is called a local ring if R has only one maximal ideal. This is equivalent to saying that the sum of two nonunits is a non-unit. Therefore the theory of all commutative local rings is axiomatizible by a finite set of A2-sentences. A commutative local ring with identity is said to be an algebraically closed local ring if every finite system of polynomial equations and inequations in one or more variables with coefficients in R which has a solution in some commutative local extension of R already has a solution in R. Much work connected with algebraically closed structures of classes of rings has been done, for example by Cherlin [2], Macintyre [4] and Lipschitz and Saracino [3]. We want to show similar results for commutative local rings with identity. Our main results are the following:Theorem. The theory of commutative local rings with identity has no model-companion.The finitely generic and infinitely generic local rings are algebraically closed local rings.Theorem. There is an A3 sentence which holds for all finitely generic local rings whose negation holds in every infinitely generic local ring.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

A commutative ring R with identity is called indecomposable if R has only the trivial idempotents, i.e. in R holds: V Vo(V 2 = Vo---~ v0 = 0 v v0 = 1). Therefore the theory of all commutative indecomposable rings with identity is axiomatizable by a finite set of universal sentences. A commutative indecomposable ring with identity is said to be an algebraically closed indecomposable ring if every finite system of polynomial equations and inequations in one or more variables with coefficients in R which has a solution in some commutative indecomposabte extension of R already has a solution in R. We will show in the present paper results for commutative indecomposable rings with identity similar to those proved by Cherlin for commutative rings (2) and by us (see [7]) for commutative local rings. Our main results are the following:

A map f : R → S {f\colon R\to S} between (associative, unital, but not necessarily commutative) rings is a brachymorphism if f ⁢ ( 1 + x ) = 1 + f ⁢ ( x ) {f(1+x)=1+f(x)} and f ⁢ ( x ⁢ y ) = f ⁢ ( x ) ⁢ f ⁢ ( y ) {f(xy)=f(x)f(y)} whenever x , y ∈ R {x,y\in R} . We tackle the problem whether every brachymorphism is additive (i.e., f ⁢ ( x + y ) = f ⁢ ( x ) + f ⁢ ( y ) {f(x+y)=f(x)+f(y)} ), showing that in many contexts, including the following, the answer is positive: • R is finite (or, more generally, R is left or right Artinian); • R is any ring of 2 × 2 {2\times 2} matrices over a commutative ring; • R is Engelian; • every element of R is a sum of π-regular and central elements (this applies to π-regular rings, Banach algebras, and power series rings); • R is the full matrix ring of order greater than 1 over any ring; • R is the monoid ring K ⁢ [ M ] {K[M]} for a commutative ring K and a π-regular monoid M; • R is the Weyl algebra 𝖠 1 ⁢ ( K ) {\mathsf{A}_{1}(K)} over a commutative ring K with positive characteristic; • f is the power function x ↦ x n {x\mapsto x^{n}} over any ring; • f is the determinant function over any ring R of n × n {n\times n} matrices, with n ≥ 3 {n\geq 3} , over a commutative ring, such that if n > 3 {n>3} , then R contains n scalar matrices with non zero divisor differences.

A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let Gc* be the subgraph of G induced on the vertex set V (G)\ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G = Γ(R), which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that Gc* has at least two connected components. We prove that the diameter of the induced graph Gc* is two if Z(R)2 ≠ {0}, Z(R)3 = {0} and Gc* is connected. We determine the structure of R which has two distinct nonadjacent vertices α, β ∈ Z(R)* \ {c} such that the ideal [N(α) ∩ N(β)]∪ {0} is generated by only one element of Z(R)*\{c}. We also completely determine the correspondence between commutative rings and finite complete graphs Kn with some end vertices adjacent to a single vertex of Kn.

Let R be a commutative ring with $$1\ne 0$$ and the additive group $$R^+$$ . Several graphs on R have been introduced by many authors, among zero-divisor graph $$\Gamma _1(R)$$ , co-maximal graph $$\Gamma _2(R)$$ , annihilator graph AG(R), total graph $$ T(\Gamma (R))$$ , cozero-divisors graph $$\Gamma _\mathrm{c}(R)$$ , equivalence classes graph $$\Gamma _\mathrm{E}(R)$$ and the Cayley graph $$\mathrm{Cay}(R^+ ,Z^*(R))$$ . Shekarriz et al. (J. Commun. Algebra, 40 (2012) 2798–2807) gave some conditions under which total graph is isomorphic to $$\mathrm{Cay}(R^+ ,Z^*(R))$$ . Badawi (J. Commun. Algebra, 42 (2014) 108–121) showed that when R is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if R has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results, we prove that for a commutative finite ring R with $$|\mathrm{Max}(R)|=n \ge 3$$ , $$ \Gamma _1(R) \simeq \Gamma _2(R)$$ if and only if $$R\simeq \mathbb {Z}^n_2$$ ; if and only if $$\Gamma _1(R) \simeq \Gamma _\mathrm{E}(R)$$ . Also the annihilator graph is identical to the cozero-divisor graph if and only if R is a Frobenius ring.

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.”

The complexity of equivalence for commutative rings

This paper is mainly concerned with describing complete theories of modules by decomposing them (up to elementary equivalence) into direct products of simpler modules. In §1, I give a decomposition theorem which works for arbitrary direct product theories T. Given such a T, I define T-indecomposable structures and show that every model of T is elementarily equivalent to a direct product of T-indecomposable models of T. In §2, I show that if R is a commutative ring then every R-module is elementarily equivalent to ΠMM where M ranges over the maximal ideals of R and M is the localization of at M. This is applied to prove that if R is a commutative von Neumann regular ring and TR is the theory of R-modules then the TR-indecomposables are precisely the cyclic modules of the form R/M where M is a maximal ideal. In §3, I use the decomposition established in §2 to characterize the ω1-categorical and ω-stable modules over a countable commutative von Neumann regular ring and the superstable modules over a commutative von Neumann regular ring of arbitrary cardinality. In the process, I also prove several general characterizations of ω-stable and superstable modules; e.g., if R is any countable ring, then an R-moduIe is ω-stable if and only if every R-module elementarily equivalent to it is equationally compact.

Let [Formula: see text] be a commutative ring with unity. The cozero-divisor graph of [Formula: see text] denoted by [Formula: see text] is a graph with the vertex set [Formula: see text], where [Formula: see text] is the set of all nonzero and non-unit elements of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote the clique number and the chromatic number of [Formula: see text], respectively. In this paper, we prove that if [Formula: see text] is a finite commutative ring, then [Formula: see text] is perfect. Also, we prove that if [Formula: see text] is a commutative Artinian non-local ring and [Formula: see text] is finite, then [Formula: see text]. For Artinian local ring, we obtain an upper bound for the chromatic number of cozero-divisor graph. Among other results, we prove that if [Formula: see text] is a commutative ring, then [Formula: see text] is a complete bipartite graph if and only if [Formula: see text], where [Formula: see text] and [Formula: see text] are fields. Moreover, we present some results on the complete [Formula: see text]-partite cozero-divisor graphs.

Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. When they are defined over finite fields, those codes have been intensively studied, especially in recent years, but they have been firstly partially characterized by Ashikhmin and Barg since 1998. Next, they were completely characterized in 2018 by Ding, Heng, and Zhou in terms of the minimum and maximum nonzero weights in the corresponding codes. Since then, many construction methods for minimal linear codes over finite fields throughout algebraic and geometric approaches have been proposed in the literature. In particular, the algebraic approach gives rise to minimal codes from (cryptographic) functions. Linear codes over finite fields have been expanded into the collection of acceptable alphabets for codes and study codes over finite commutative rings. A natural way to extend the known results available in the literature is to consider minimal linear codes over commutative rings with unity. In extending coding theory to codes over rings, several essential principles must be considered. Particularly extending the minimality property from finite fields to rings and creating such codes is not simple. Such an extension offers more flexibility in the construction of minimal codes. The present article investigates one-dimensional minimal linear codes over the rings <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}_{p^{n}}$ </tex-math></inline-formula> (where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> is a prime) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}_{p^{m}q^{n}}$ </tex-math></inline-formula> (where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p &lt; q$ </tex-math></inline-formula> are distinct primes and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\leq n$ </tex-math></inline-formula> ). Our ultimate objective is to characterize such codes’ minimality and design minimal linear codes over the considered rings. Given our objective, we first introduced the notion of minimal codes over (commutative) rings and succeeded in deriving simple characterization of one-dimensional minimal linear codes over the underlying rings mentioned above. Our new algebraic approach allows designing new minimal linear codes. Almost minimal codes over rings are also presented. To the best of our knowledge, the present paper offers a wide variety of minimal codes over (commutative) rings for the first time. Novel perspectives and developments in this direction are expected in the future.

Unique decomposition into ideals for commutative Noetherian rings

Commutative Noetherian local rings whose ideals are direct sums of cyclic modules