On Rings in Which Every Ideal Is Weakly Prime

Abstract

Abstract. Anderson-Smith [1] studied weakly prime ideals for a commutative ring with identity. Blair-Tsutsui [2] studied the structure of a ring in which every ideal is prime. In this paper we investigate the structure of rings, not necessarily commutative, in which all ideals are weakly prime.

Weakly Prime and Weakly Completely Prime Ideals of Noncommutative Rings

“…Anderson-Smith [1] defined a proper ideal P of a commutative ring R with identity to be weakly prime if 0 = ab ∈ P implies a ∈ P or b ∈ P . In [5] Hirano et al…”

“…In this paper, all rings are, as in [5], not necessarily commutative nor with identity. By a ring R with identity, we shall mean that R has a multiplicative identity 1 = 0.…”

“…Equivalent to this is that if a, b ∈ R such that aRb ⊆ P then a ∈ P or b ∈ P . From [5] we have the following: Proposition 1.1. [5, Proposition 2] Let P be an ideal in a ring R with identity.…”

Characterization of weakly primary ideals over non-commutative rings

“…Atani and F. Farzalipour [2]. They defined a weakly primary ideal as a proper ideal P over a commutative ring R with the property that if 0 ̸ = ab ∈ P, then a ∈ P or b n ∈ P for some positive integer n. The structure of weakly prime ideals over non-commutative rings has been studied by Y. Hirano, E. Poon, and H. Tsutsui in [6]. Also, they investigated the structure of rings, not necessarily commutative nor with identity, in which all ideals are weakly prime.…”

“…Various properties of weakly primary (weakly prime) subtractive ideals over commutative semirigs have been studied in [3] and [5]. The motivation of this paper is to continue the studying of the family of primary ideals, also to extend the results of Anderson [1], Atani and Frazalipour [2], and Hirano, Poon, and Tsutsui [6] to the weakly primary ideals over noncommutative rings.…”