A Connection Between Weak Regularity and the Simplicity of Prime Factor Rings

Abstract. Let R be a ring. It is well-known that R is NI if and only if n i=0 Ra i R is a nil ideal of R whenever a polynomial n i=0 a i x i is nilpotent, where x is an indeterminate over R. We consider a condition which is similar to the preceding one: n i=0 Ra i R contains a nonzero nil ideal of R whenever n i=0 a i x i over R is nilpotent. A ring will be said to be quasi-NI if it satisfies this condition. The structure of quasi-NI rings is observed, and various examples are given to situations which raised naturally in the process.

show abstract

“…Moreover R\N * (R) is correspondent to Z\{0}, so every element in R\N * (R) is regular in R. Thus M at n (R) is quasi-NI by Proposition 1.5 (1).…”

Section: Quasi-ni Ringsmentioning

confidence: 86%

“…Let A be a division ring and R = U n (A) or R = D n (A) for n ≥ 2. Then R is π-regular by [1,Corollary 6], and R is not regular by the existence of nonzero N * (R). Moreover R is quasi-NI by Proposition 2.1, but R is not reduced.…”

Section: About Ordinary Ring Extensionsmentioning

confidence: 99%

On Ni and Quasi-Ni Rings

Kim

Lee

et al. 2016

Korean Journal of Mathematics

View full text Add to dashboard Cite

show abstract

“…Indeed, D n (A) is π-regular by [5,Corollary 6], but it is clearly not regular when n ≥ 2. There exist many non-Abelian π-regular rings by [5,Corollary 6], as can be seen by U n (K) over a division ring K for n ≥ 2.…”

Section: Relations and Examplesmentioning

confidence: 99%

Structures Concerning Group of Units

Chung¹,

Lee²

2017

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. In this note we consider the right unit-duo ring property on the powers of elements, and introduce the concept of weakly right unitduo ring. We investigate first the properties of weakly right unit-duo rings which are useful to the study of related studies. We observe next various kinds of relations and examples of weakly right unit-duo rings which do roles in ring theory. Basic structure of weakly right unit-duo ringsThroughout this paper all rings are associative with identity unless otherwise stated. Let R be a ring and a ∈ R. We use U (R) to denote the group of all units in R., and N (R) denote the Jacobson radical, the set of all nonzero nonunits, the set of all idempotents, and the set of all nilpotent elements in R, respectively. |S| denotes the cardinality of a subset S of R. The polynomial (power series) ring, with an indetermainate x over R, is written by. Z (Z n ) denotes the ring of integers (modulo n), and Q denotes the field of rational numbers. Denote the n by n full (resp., upper triangular) matrix ring over R by Mat n (R) (resp., U n (R)), and use E ij for the matrix with (i, j)-entry 1 and elsewhere zeros. Following the literature, we write D n (R) = {(a ij ) ∈ U n (R) | a 11 = · · · = a nn } and R * = R\{0}. We use ⊕ to denote the direct sum, and Q 8 denotes the quaternion group. Due to Feller [11], a ring is called right (resp. left) duo if every right (resp. left) ideal is an ideal; a ring is called duo if it is both right and left duo. We see various kinds of useful results for duo rings in [6,21,26]

show abstract

“…Since N(R) is completely semiprime, Lemma 7 of [5] yields a"s\ • • • s k e N(7? ), where n = MI + h M/t.…”

Section: Let R Be a 2-primal Ring And P A Prime Ideal Of R (I) Ifr Smentioning

confidence: 99%

“…If xRs ^ 0, then there exists r e R such that xrs ^ 0. Since P(R) is completely semiprime, Lemma 7 of [5] yields xrs e P(R). Therefore O(P) + P(R) is right essential in P. LEMMA …”

Section: Let R Be a 2-primal Ring And P A Prime Ideal Of R Then: (I)mentioning

confidence: 99%

A characterization of minimal prime ideals

1998

Self Cite

View full text Add to dashboard Cite

Abstract. Let P be a prime ideal of a ring R, O(P) = [a € R\ aRs = 0, for some .? € R\P] and O(P) -{x e R | x" e O(P), for some positive integer n). Several authors have obtained sheaf representations of rings whose stalks are of the form R/O{P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P = O(P). In this paper we derive various conditions which ensure that a prime ideal P -O(P). The property that P = 0{P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and O(P) are considered. Examples are provided to illustrate and delimit our results. A proper ideal P of R is called completely prime (completely semiprime) if xy e P (x 2 e P) implies x e P or y e P (x e P). Andrunakievic and Rjabuhin [1] and , independently, Stewart[18] have shown that a reduced ring R (i.e., N(/?) = 0) is a subdirect product of integral domains. Thus a proper ideal is completely semiprime if and only if it is an intersection of completely prime ideals.All prime ideals are taken to be proper ideals. Let A' be a nonempty subset of R, then R , 1{X) and r(X) denote the ideal of R generated by X, the left annihilator of X in R, and the right annihilator of X in R, respectively. Let P be a prime ideal. The following definitions are fundamental to the remainder of our discussion:

show abstract

A Connection Between Weak Regularity and the Simplicity of Prime Factor Rings

Cited by 13 publications

References 3 publications

On Ni and Quasi-Ni Rings

On Ni and Quasi-Ni Rings

Structures Concerning Group of Units

A characterization of minimal prime ideals

Contact Info

Product

Resources

About