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The First Isomorphism Theorem and Other Properties of Rings

Artur Korniłowicz, Christoph Schwarzweller (2014)

Formalized Mathematics

Different properties of rings and fields are discussed [12], [41] and [17]. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism f : R → S we have R/ker(f) ≅ Im(f). Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and hence also factorial

The maximal regular ideal of some commutative rings

Emad Abu Osba, Melvin Henriksen, Osama Alkam, Frank A. Smith (2006)

Commentationes Mathematicae Universitatis Carolinae

In 1950 in volume 1 of Proc. Amer. Math. Soc., B. Brown and N. McCoy showed that every (not necessarily commutative) ring R has an ideal 𝔐 ( R ) consisting of elements a for which there is an x such that a x a = a , and maximal with respect to this property. Considering only the case when R is commutative and has an identity element, it is often not easy to determine when 𝔐 ( R ) is not just the zero ideal. We determine when this happens in a number of cases: Namely when at least one of a or 1 - a has a von Neumann inverse,...

The ∘ operation and * operation of Cohen-Macaulay bipartite graphs

Yulong Yang, Guangjun Zhu, Yijun Cui, Shiya Duan (2024)

Czechoslovak Mathematical Journal

Let G be a finite simple graph with the vertex set V and let I G be its edge ideal in the polynomial ring S = 𝕂 [ V ] . We compute the depth and the Castelnuovo-Mumford regularity of S / I G when G = G 1 ∘ G 2 or G = G 1 * G 2 is a graph obtained from Cohen-Macaulay bipartite graphs G 1 , G 2 by the ∘ operation or * operation, respectively.

The prime ideals intersection graph of a ring

M. J. Nikmehr, B. Soleymanzadeh (2017)

Commentationes Mathematicae Universitatis Carolinae

Let R be a commutative ring with unity and U ( R ) be the set of unit elements of R . In this paper, we introduce and investigate some properties of a new kind of graph on the ring R , namely, the prime ideals intersection graph of R , denoted by G p ( R ) . The G p ( R ) is a graph with vertex set R * - U ( R ) and two distinct vertices a and b are adjacent if and only if there exists a prime ideal 𝔭 of R such that a , b ∈ 𝔭 . We obtain necessary and sufficient conditions on R such that G p ( R ) is disconnected. We find the diameter and girth of G p ( R ) ....

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