Weakly prime ideals.(English)Zbl 1086.13500

Summary: Let $$R$$ be a commutative ring with identity. We define a proper ideal $$P$$ of $$R$$ to be weakly prime if whenever $$0\neq ab\in P$$ then either $$a\in P$$ or $$b\in P$$. For example, every proper ideal of a quasilocal ring $$(R,M)$$ with $$M^2=0$$ is weakly prime. We show that a weakly prime ideal $$P$$ that is not prime satisfies $$P^2=0$$, in fact, $$P$$nil($$R)= 0$$. A number of results concerning weakly prime ideals and examples of weakly prime ideals are given. We show that every proper (principal) ideal of $$R$$ is a product of weakly prime ideals if and only if $$R$$ is a finite direct product of Dedekind domains (locally factorial Krull domains) and SPIR’s or $$(R,M)$$ is a quasilocal ring with $$M^2=0$$.

MSC:

 13A15 Ideals and multiplicative ideal theory in commutative rings 13H99 Local rings and semilocal rings