A Concept Unifying the Armendariz and Ni Conditions

Abstract

Abstract. We study the structure of the set of nilpotent elements in various kinds of ring and introduce the concept of NR ring as a generalization of Armendariz rings and N I rings. We determine the precise relationships between N R rings and related ring-theoretic conditions. The Köthe's conjecture is true for the class of N R rings. We examined whether several kinds of extensions preserve the N R condition. The classical right quotient ring of an N R ring is also studied under some conditions on the subset of nilpotent elements.

ON RINGS IN WHICH ALL UNITS CAN BE PRESENTED IN THE FORM 1+eR(1-e)

“…If the set of nilpotent elements of a ring is a subring, then the ring is called an 𝑁𝑅-ring (see [1]).…”

“…In this study, some comparisons between 𝑁𝑅-rings, 𝑈𝑈-rings, 𝑈𝐽-rings, 𝑈𝑅-rings, exchange rings, 2primal rings, are investigated. We show that if 𝑅 is a ring in which 1 + 𝑒𝑅(1 − 𝑒) = 𝑈(𝑅), then 𝑅 is a 𝑈𝐽-ring (if every unit can be presented in a form 1 + 𝑥, for some 𝑥 ∈ 𝐽(𝑅) [5] (see also [6] and [11])) and 𝑅 is an 𝑁𝑅-ring (if the set of nilpotent elements of a ring is a subring [1]). In particular, we obtain that ring for which 1 + 𝑒𝑅(1 − 𝑒) = 𝑈(𝑅) satisfy Köthe's conjecture.…”

“…10 (1) -2022 Let 𝑅 be a ring in which 1 + 𝑒𝑅(1 − 𝑒) = 𝑈(𝑅) for some 𝑒 2 = 𝑒 ∈ 𝑅. Then:(1) 𝑒 = 𝑢𝑒 for every 𝑢 ∈ 𝑈(𝑅) (2) (1 − 𝑒) = (1 − 𝑒)𝑢 for every 𝑢 ∈ 𝑈(𝑅) (3) 𝑒𝑈(𝑅)𝑒 = 𝑒 (4) (1 − 𝑒)𝑈(𝑅)(1 − 𝑒) = (1 − 𝑒). (5) (1 − 𝑒)𝑅𝑒 = 0 (6) 𝑒𝑅𝑒 = 𝑅𝑒 (7) (1 − 𝑒)𝑅(1 − 𝑒) = (1 − 𝑒)𝑅.It is clear.…”

Amalgamated rings with clean-type properties

“…Recall by [8], that a ring R forms an NR ring if nil(R) is a subring of R. In [2], Antoin constructs various examples of Armendariz and related rings by reviewing and extending some results concerning the structure of nil(R). Anyway, the set of nilpotent elements forms a subring without unit, and this is true not only in the case of R Armendariz, but in many of the generalizations of this condition.…”

Quasi-normality of idempotents on nilpotents

“…From Proposition 3.7, we can obtain an information for matrix rings to be right IQNN. Following [6], a ring R is called NR if N (R) forms a subring of R. NI rings are clearly NR, but the converse need not hold by Antoine [2,Example 4.8]. We first observe the NR property of the rings in Theorem 2.4 and Example 2.5 for our purpose.…”