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 \(M^2=0\).


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