1. Structure of zero-divisors to go up to related ideals.
- Author
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Jung, Da Woon, Lee, Chang Ik, Lee, Yang, and Piao, Zhelin
- Subjects
- *
MATRIX rings , *POLYNOMIAL rings , *POLYNOMIALS , *RING theory - Abstract
AbstractThere are many ways for zero-dividing polynomials to go up to zero-dividing ideals when the base rings are IFP. The importance of these in ring theory leads us to consider the following ring conditions and study new useful roles of matrices for ring theory. Let
R be a ring and a,b∈R\{0}. The first is the condition (*) that ifab = 0 thenIb = 0 for some nonzero ideal I⊆RaR ofR oraJ = 0 for some nonzero ideal J⊆RbR ofR . It is shown that from given any IFP ring, there can be constructed a non-IFP ring with the condition (*). We prove that a semiprime ringR with the condition (*) is both right and left nonsingular. The second is the condition (**) that ifab = 0 thenIJ = 0 for some nonzero ideals I⊆RaR and J⊆RbR ofR . We prove that every ring can be a subring of rings with the condition (**), that ifR is an irreducible ring with the condition (**) thenR is either a domain or non-semiprime, and that the condition (**) passes to polynomial rings when the base ring is semiprime. Various sorts of examples are given to illustrate and delimit the results obtained. [ABSTRACT FROM AUTHOR]- Published
- 2024
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