The Prime Avoidance Lemma Revisited

Abstract

Abstract. We show that the above lemma and its well-known refinement are valid, in a general setting, in non-commutative rings. Some interesting consequences are also observed.The above lemma is in every single text book on commutative algebra, but not even a single book on non-commutative algebra, as yet (except [12, Proposition 2.12.7], where the validity of a special variant of this lemma is shown), proves it for non-commutative rings. We recall that, in the proof of [7, Proposition 2], which is in fact the proof of the converse of the Generalized Principal Ideal Theorem of Krull for non-commutative rings, see also [5, Theorem 153], we invoked the lemma for the non-commutative case, even without mentioning it, and gave no proof for it ( note, it seems (at least to us) that the lemma for non-commutative rings was overlooked in the literature, at that time, and although I knew of a proof for a generalization of this lemma then, but did not present it in [7], on purpose, for the reason that we will see, shortly). Let R be a ring with identity, if S is a subring of R without containing the identity of R, we say that S is a subring−1 (see [4, Chapter 16]). The above lemma states that, if P 1 , P 2 , . . . , P n are ideals of a commutative ring R with identity such that at most two of the P i 's are not prime and S is a subring−1 of R contained in P 1 ∪ P 2 ∪ . . . ∪ P n , then S ⊆ P k for some P k (note, in most cases S is an ideal ). Without any doubt, as it is rightly mentioned in [13], the Prime Avoidance Lemma is one of the fundamental cornerstones of commutative algebra. Its numerous applications in the field, makes one surely claim that no one working in the commutative algebra can do without it. The lemma for commutative rings goes back to a 1957 paper by McCoy, and Kaplansky generalized it in his 1974 book on commutative rings, see [4], [5], respectively. The reader might consult ([4, Chapter 16]), [9] and [11]) for a short history and some variations of this lemma. There is also a very useful refinement of the lemma, which says that whenever P 1 , P 2 , . . . , P n are prime ideals of the commutative ring R and I is an ideal of R, a ∈ R with

$r$-ideals in commutative rings

“…To see this, we prove in fact that each maximal ideal is a minimal prime ideal. Let M ∈ Max(Q(R)); since Q(R) is a uz -ring, we have M ⊆ zd(Q(R)) , so M is a The following proposition is a counterpart of the celebrated Prime Avoidance Lemma for r -ideals; see [ [21]] for recent work on this lemma. First we need the next definition.…”

“…The following proposition is a counterpart of the celebrated Prime Avoidance Lemma for r -ideals; see [ [21]] for recent work on this lemma. First we need the next definition.…”

Uniformly Primal Submodule over Noncommutative Ring

“…en, RaRbR ⊆ ∪ n i�1 P i . Hence, by the prime avoidance theorem for rings (see [3] and [4]), we have (RaR)(RbR) � RaRbR ⊆ P i for some i � 1, 2, . .…”

“…, n { }. Karamzadeh [4] generalizes the prime avoidance theorem for any ring that is not necessarily commutative. e aim of Section 1 is to generalize the prime avoidance theorem for rings over noncommutative rings to the uniformly primal avoidance theorem over noncommutative rings.…”

The Space of Maximal Subrings of a Commutative Ring

“…Also we refer the reader to see [14,26,30,35], for more observations about this lemma. The following corollary is the counterpart for valuations.…”