In this paper, we prove that R is a two-sided Artinian ring and J is a rightannihilator ideal if and only if (ⅰ) for any nonzero right module, there is a nonzero linear mapfrom it to a projective module; (ⅱ) every s...In this paper, we prove that R is a two-sided Artinian ring and J is a rightannihilator ideal if and only if (ⅰ) for any nonzero right module, there is a nonzero linear mapfrom it to a projective module; (ⅱ) every submodule of RR is not a radical module for some rightcoherent rings. We call a ring a right X ring if Homfl(M, R) = 0 for any right module M implies thatM = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover wecharacterize semisimple rings by using X rings. A famous Faith's conjecture is whether a semipimaryPF ring is a QF ring. Similarly we study the relationship between X rings and QF and get manyinteresting results.展开更多
Suppose R be a commutative ring with an identity element 1. Tong Wenting hasstudied PF rings in ref.[1]. A ring R is called a PF ring if every finitely generatedprojective R-module is free. In this note, we construct ...Suppose R be a commutative ring with an identity element 1. Tong Wenting hasstudied PF rings in ref.[1]. A ring R is called a PF ring if every finitely generatedprojective R-module is free. In this note, we construct a new abelian group X(R). Asa ring, we will prove X(R) is a PF ring.展开更多
文摘In this paper, we prove that R is a two-sided Artinian ring and J is a rightannihilator ideal if and only if (ⅰ) for any nonzero right module, there is a nonzero linear mapfrom it to a projective module; (ⅱ) every submodule of RR is not a radical module for some rightcoherent rings. We call a ring a right X ring if Homfl(M, R) = 0 for any right module M implies thatM = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover wecharacterize semisimple rings by using X rings. A famous Faith's conjecture is whether a semipimaryPF ring is a QF ring. Similarly we study the relationship between X rings and QF and get manyinteresting results.
文摘Suppose R be a commutative ring with an identity element 1. Tong Wenting hasstudied PF rings in ref.[1]. A ring R is called a PF ring if every finitely generatedprojective R-module is free. In this note, we construct a new abelian group X(R). Asa ring, we will prove X(R) is a PF ring.
