Commutative rings with two-absorbing factorization

Factorization in Commutative Rings

MATH 436 Notes: Factorization in Commutative Rings

Proposition 1.1. Let f : R1 → R2 be a homomorphism of rings. If J is an ideal of R2, then f (J) is an ideal of R1 containing ker(f) and furthermore f(f(J)) ⊆ J . Now let f : R1 → R2 be an epimorphism of rings. If J is an ideal of R2 then f(f (J)) = J . If I is an ideal of R1 then f(I) is an ideal of R2. Furthermore we have I ⊆ f (f(I)) = I + ker(f) and thus I = f(f(I)) if I contains ker(f). Thu...

On 2-absorbing Primary Submodules of Modules over Commutative Rings

All rings are commutative with 1 6= 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a, b ∈ R and m ∈M and abm ∈ N , then am ∈M -rad(N) or bm ∈M -rad(N) or ...

Results on n-Absorbing Ideals of Commutative Rings

RESULTS ON N-ABSORBING IDEALS OF COMMUTATIVE RINGS by Alison Elaine Becker The University of Wisconsin-Milwaukee, 2015 Under the Supervision of Dr. Allen Bell Let R be a commutative ring with 1 6= 0. In his paper On 2-absorbing ideals of commutative rings, Ayman Badawi introduces a generalization of prime ideals called 2-absorbing ideals, and this idea is further generalized in a paper by Ander...

Tiling with Commutative Rings

so that every square is covered by exactly one domino? In other words, can R be tiled by vertical and horizontal dominoes? The coloring gives the answer to this well-known problem away. The region R has 32 black squares and 30 white squares. Since each domino covers exactly one black and one white square, no tiling is possible. The aim of this article is to explain a way to tackle tiling proble...