$r$-ideals in commutative rings

Abstract

Turkish Journal of Mathematics h t t p : / / j o u r n a l s . t u b i t a k . g o v . t r / m a t hAbstract: In this article we introduce the concept of r -ideals in commutative rings (note: an ideal I of a ring R is called r -ideal, if ab ∈ I and Ann(a) = (0) imply that b ∈ I for each a, b ∈ R ). We study and investigate the behavior of r -ideals and compare them with other classical ideals, such as prime and maximal ideals. We also show that some known ideals such as z • -ideals are r -ideals. It is observed that if I is an r -ideal, then so too is a minimal prime ideal of I . We naturally extend the celebrated results such as Cohen's theorem for prime ideals and the Prime AvoidanceLemma to r -ideals. Consequently, we obtain interesting new facts related to the Prime Avoidance Lemma. It is also shown that R satisfies property A (note: a ring R satisfies property A if each finitely generated ideal consisting entirely of zerodivisors has a nonzero annihilator) if and only if for every r -ideal. Using this concept in the context of C(X) , we show that every r -ideal is a z • -ideal if and only if X is a ∂ -space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior). Finally, we observe that, although the socle of C(X) is never a prime ideal in C(X) , the socle of any reduced ring is always an r -ideal.

J-Ideals of Commutative Rings

“…Recently, [10] Rostam Mohamadian has introduced the concept of r-ideals in commutative rings. A proper ideal I of a ring R is called an r-ideal if whenever a, b ∈ R with ab ∈ I and Ann (a) = 0, then b ∈ I where Ann (a) = {r ∈ R : ra = 0}.…”

On uniformlypr-ideals in commutative rings

“…The aim of this article is to introduce uniformly pr -ideals of commutative rings and to give relations with some classical ideals such as uniformly primary ideal, strongly primary ideal, r -ideal. In [12], Mohammadian introduces r -ideals of commutative rings which is the generalization of pure ideals. Recall that a proper ideal I of R is called an r -ideal if whenever ab ∈ I and ann(a) = 0 then b ∈ I (b n ∈ I, for some n ∈ N).…”

“…Recall that a proper ideal I of R is called an r -ideal if whenever ab ∈ I and ann(a) = 0 then b ∈ I (b n ∈ I, for some n ∈ N). In terms of r -ideals the author characterizes quasiregular rings, rings satisfying property A (See [12], Theorem 4.2) and ( [12], Proposition 3.5).…”

r -Submodules and sr-Submodules

“…In particular, R will always denote such a ring. The concept of r -ideals was introduced and studied by Mohamadian in [9]. Recall from [9] that a proper ideal I of R is an r -ideal if ab ∈ I and ann(a) = {r ∈ R : ra = 0} = 0 , and then b ∈ I for each a, b ∈ R .…”