S-principal ideal multiplication modules

For a commutative local ring R, we know that every ideal is a direct sum of local modules if and only if the unique maximal ideal of R is of the form , where and are principal ideal rings and for all γ’s. This motivates us to determine a commutative ring, not necessarily local, in which every (proper) ideal is a direct sum of local modules. We prove that such a ring is exactly a finite direct product of local rings R with the unique maximal ideal in the above mentioned form. Moreover, we characterize more precisely certain commutative rings such as Noetherian rings, reduced rings, Rickart rings, semi-Artinian rings and self-injective rings, in which every (proper) ideal is a direct sum of local modules.

On some Euclidean properties of matrix algebras

On lucky ideals for Gröbner basis computations

We give a negative answer to a question posed by A. V. Jategaonkar: is it not true that an arbitrary primary principal left ideal ring is a factor of a prime principal left ideal ring? We give a counter example in the class of finite complete primary principal ideal rings, the so-called Galois-Eisenstein-Ore rings.

We prove that if $R$ is a principal ideal ring and $A\in \mathrm {M}_{n}(R)$ is a matrix with trace zero, then $A$ is a commutator, that is, $A=XY-YX$ for some $X,Y\in \mathrm {M}_{n}(R)$. This generalises the corresponding result over fields due to Albert and Muckenhoupt, as well as that over $\mathbb {Z}$ due to Laffey and Reams, and as a by-product we obtain new simplified proofs of these results. We also establish a normal form for similarity classes of matrices over PIDs, generalising a result of Laffey and Reams. This normal form is a main ingredient in the proof of the result on commutators.

On the depth of the Rees algebra of an ideal module

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

The notion of weakly [Formula: see text]-absorbing ideals, introduced as a natural generalization of prime ideals, has gained significant attention in commutative rings. A weakly [Formula: see text]-absorbing ideal of a commutative ring [Formula: see text] ensures that for all [Formula: see text], the inclusion [Formula: see text] and [Formula: see text] implies that at least one of [Formula: see text] or [Formula: see text] belongs to [Formula: see text]. This paper builds upon the existing framework by introducing and exploring weakly [Formula: see text]-[Formula: see text]-absorbing ideals in the context of a multiplicatively closed subset [Formula: see text] of [Formula: see text]. An ideal [Formula: see text], disjoint from [Formula: see text], is defined as weakly [Formula: see text]-[Formula: see text]-absorbing if [Formula: see text] and [Formula: see text] guarantee that [Formula: see text], [Formula: see text], or [Formula: see text] for some [Formula: see text]. We give the fundamental properties of weakly [Formula: see text]-[Formula: see text]-absorbing ideals and investigate their behavior under various ring constructions, such as amalgamation duplications, homomorphic images, direct products and trivial extensions. Notable results include conditions under which the ideals retain the weakly [Formula: see text]-[Formula: see text]-absorbing property and characterizations of these ideals in special commutative rings such as reduced rings and local semiprime rings which are not prime.

The self-idealization of a commutative ring R is isomorphic to the ring \(R[x]/(x^2)\) or, equivalently, the ring of upper-triangular Toeplitz matrices \({{\mathrm{\mathcal {T}}}}(R)=\left\{ \left( \begin{matrix} a &{} b \\ 0 &{} a \end{matrix}\right) :a,b\in R\right\} \). Recently, Chang and Smertnig characterized the sets of lengths of factorizations in \({{\mathrm{\mathcal {T}}}}(D)\) where D is a principal ideal domain. In this work, in addition to correcting an error in their paper, we extend the study to \({{\mathrm{\mathcal {T}}}}(R)\) when R is either a principal ideal ring or a unique factorization ring.

The aim of the article is to give a characterization of a multiplication commutative ring with finitely many minimal prime ideals: Each such ring is a finite direct product of rings where Di is either a Dedekind domain or an Artinian, local, principal ideal ring and vice versa. In particular, each such ring is a Noetherian ring. As a corollary subclasses of such rings are described (semiprime, Artinian, semiprime and Artinian, local, domain, etc.).

For an element $x$ of a commutative ring $R$, let $\Gamma_x(R)$ be the graph whose vertices are the elements of $R$ that divide $x$ such that distinct vertices $r$ and $s$ are adjacent if and only if $rs=x$. If $x$ is a nonzero nonirreducible nonunit of an integral domain $R$, then $\Gamma_x^\mathcal{C}(R)$ is the graph whose vertices are the associate classes of divisors of $x$ that are neither units nor associates of $x$ such that distinct vertices $A$ and $B$ are adjacent if and only if $rs$ divides $x$ for some (and hence every) $r\in A$ and $s\in B$. The graphs $\Gamma_x(R)$ are considered when $R$ is a local Artinian principal ideal ring, and $\Gamma_x^\mathcal{C}(R)$ is examined when $R$ is a valuation domain. For example, it is shown that a finite local ring $R$ is a principal ideal ring if and only if there exists $x\in R$ such that every connected component of $\Gamma_x(R)$ is a star graph of a prescribed cardinality. Moreover, it is proved that an integral domain $R$ is a discrete valuation ring if and only if its collection of graphs $\Gamma_x^\mathcal{C}(R)$ consists precisely of a single-vertex graph, along with every (up to isomorphism) graph that is realizable as the compressed $0$-divisor graph of a local Artinian principal ideal ring. Certain graphs associated with partially ordered abelian groups have an essential role in the work.

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and S that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4].

We study the structure of cyclic codes of length k 2 over 8 Z for any natural number k. It is known that cyclic codes , thereby implying that R is not a principal ideal ring. We also prove that cyclic codes of length k 2 over 8 Z are generated as ideals by at most three elements.

In this paper, we considered linear block codes over Rq=Fq+uFq+vFq+uvFq, u2=v2=0,uv=vu where q=pm, m∈N . First we looked at the structure of the ring. It was shown that Rq is neither a finite chain ring nor a principal ideal ring but is a local ring. We then established a generator matrix for the linear block codes and equipped it with a homogeneous weight function. Field codes were then constructed as images of these codes by using a basis of Rq over Fq . Bounds on the minimum Hamming distance of the image codes were then derived. A code meeting such bounds is given as an example.

Efficient Gröbner bases computation over principal ideal rings