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Let and be commutative rings with identity, be an ideal of , be a ring homomorphism, be an -module, be an -module, and let be an -homomorphism. The amalgamation of with along with respect to denoted by was introduced by M. D’Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of -module called the amalgamation of and along with respect to , and denoted by . We study some homological properties of the -module . Among other results,...
In analogy with the notion of the composite semi-valuations, we define the composite -valuation from two other -valuations and . We consider a lexicographically exact sequence and the composite -valuation of a field with value group . If the assigned to set or non comparable to is a local ring, then a -valuation of into is defined with its assigned set a local ring, as well as another -valuation of a residue field is defined with -value group .
Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper we will obtain some results concerning the graded comultiplication modules over a commutative graded ring.
The Dedekind-Mertens lemma relates the contents of two polynomials and the content of their product. Recently, Epstein and Shapiro extended this lemma to the case of power series. We review the problem with a special emphasis on the case of power series, give an answer to a question posed by Epstein-Shapiro and investigate extensions of some related results. This note is of expository character and discusses the history of the problem, some examples and announces some new results.
We introduce and study a new class of ring extensions based on a new formula involving the heights of their primes. We compare them with the classical altitude inequality and altitude formula, and we give another characterization of locally Jaffard domains, and domains satisfying absolutely the altitude inequality (resp., the altitude formula). Then we study the extensions R ⊆ S where R satisfies the corresponding condition with respect to S (Definition 3.1). This leads to a new characterization...
Let be a commutative ring. The annihilator graph of , denoted by , is the undirected graph with all nonzero zero-divisors of as vertex set, and two distinct vertices and are adjacent if and only if , where for , . In this paper, we characterize all finite commutative rings with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings whose annihilator graphs have clique number , or . Also, we investigate some properties of the annihilator...
Let k be a field and k[x,y] the polynomial ring in two variables over k. Let D be a higher k-derivation on k[x,y] and D̅ the extension of D on k(x,y). We prove that if the kernel of D is not equal to k, then the kernel of D̅ is equal to the quotient field of the kernel of D.
On the ring of polynomials in n variables over a field special isomorphisms ’s of into are defined which preserve the greatest common divisor of two polynomials. The ring is extended to the ring and the ring of generalized polynomials in such a way that the exponents of the variables are non-negative rational numbers and rational numbers, respectively. The isomorphisms ’s are extended to automorphisms ’s of the ring . Using the property that the isomorphisms ’s preserve GCD it...
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