Π-regular Rings Research Articles

On Strongly ?-Regular Group Rings

Let R be an associative ring with unit. An element x ∈ R is said to be left (right) π-regular if there exist y ∈ R and a positive integer n such that $$x^{n}=yx^{n+1}(x^{n}=x^{n+1}y)$$ . If x is both left and right π-regular, then it is said to be strongly π-regular. R is said to be a strongly π-regular ring if all its elements are strongly π-regular. In this paper we determine some conditions which are necessary or sufficient for a group ring to be strongly π-regular.

On Related Power Comparability of Modules

In this paper, we introduce the concept of generalized cu-rings, study the related power comparability of modules. Let R be an exchange ring, we show that the following are equivalent: (1) For any R-modules A and B, R R ⊕ A ≅ R R ⊕ B implies that there exist a positive integer n and an idempotent e ∈ B(R) such that (2) Given any R-module decompositions M = A 1 ⊕ B 1 = A 2 ⊕ B 2 with A 1 ≅ R R ≅ A 2, there exist n ≥ 1 and C, D, E ≤ M n such that where D is a direct summad of , E is a direct summand of and De = 0, E(1 − e) = 0 for some e ∈ B(R). (3) For any idempotents e, f ∈ R with e = 1 + ab and f = 1 + ba for some a, b ∈ R, there exist u ∈ B(R) and n ≥ 1 such that (4) For any regular element a ∈ R, there exists some n ≥ 1 such that aI n is related unit regular in M n (R). Also, we show that every regular related unit π-regular ring is a generalized cu-ring. These generalize the corresponding results in (1), (2), (3) and (4).

Semiregular, Weakly Regular, and Π-Regular Rings

Publisher Summary This chapter discusses the semi-regular, weakly regular and π-regular rings. All rings are associative and have nonzero identity elements. Every strongly π-regular ring is π-regular. The factor ring of the integers with respect to the ideal generated by the integer four is a strongly re-regular ring which is not a regular ring. For a ring A, the Jacobson radical of A is denoted by J (A) which is equal to the intersection of all maximal right ideals of A by definition. A ring A is said to be semi-regular if A / J (A) is a regular ring, and all idempotents of A / J (A) are images of idempotents of A for the natural epimorphism A → A / J (A). The classes of semi-regular and π-regular rings are quite large. Every regular ring is a semi-regular π-regular ring. All right or left Artinian rings are semi-regular π-regular rings, and the endomorphism ring of any injective module is a semi-regular ring.

RINGS AND MODULES WITH EXCHANGE PROPERTIES

All rings are assumed to be associative and (except for nil-rings and for some stipulated cases) to have nonzero identity elements. A ring A is an exchange ring if the following two equivalent conditions hold: (1) for any element a ∈ A, there exists an idempotent e ∈ aA with 1 − e ∈ (1 − a)A; (2) for any element a ∈ A, there exists an idempotent e ∈ Aa with 1− e ∈ A(1− a). (The equivalence (1)⇐⇒(2) is proved in 2.1 below.) A ring A is said to be regular if for every element a of A, there exists an element b of A such that a = aba. A ring A is semiregular if A/J(A) is a regular ring and all idempotents of A/J(A) can be lifted to idempotents of A. A ring A is a π-regular ring if, for every element a of A, there exists an element b of A such that a = aba for some positive integer n. Every semiregular or π-regular ring is an exchange ring (see 2.11(1)). It is directly verified that the ring of all rational numbers with odd denominators is an exchange ring that is not a π-regular ring. Let B be the ring of all rational numbers with odd denominators, {Ai}i=1 be a countable infinite set of copies of the field of rational numbers, D be the direct product of all rings Ai, and R be the subring in D generated by the ideal ⊕i=1Ai and by the subring {(b, b, b, . . . ) | b ∈ B}. Then R is a commutative reduced semiprimitive exchange ring that is not a semiregular ring (see 2.9(4)). A moduleM has the finite exchange property if for every moduleX and for every direct decomposition X = M ′ ⊕ Y = ⊕i∈INi, where M ′ ∼= M and I is a finite set, there exist submodules N ′ i ⊆ Ni (i ∈ I) such that X = M ′ ⊕ (⊕i∈IN ′ i). The importance of exchange rings is related to the fact that a module M has the finite exchange property if and only if the ring End(M) is an exchange ring [75]. The systematic study of modules with the finite exchange property and exchange rings was initiated in [18,58,75]. Also, see [2–4,14,17,39,43,48,55,56,58–61,63,67,73–75,77–79,82–85,87–90,93]. Exchange rings are potent rings (a ring A is potent if every right ideal of A that is not contained in J(A) contains a nonzero idempotent and all idempotents of A/J(A) can be lifted to idempotents of A). Thus, exchange rings have many idempotents. For a module M , the Jacobson radical, the endomorphism ring, and the lattice of all submodules are denoted by J(M), End(M), and Lat(M), respectively. For a ring A, C(A) and U(A) denote the center and the group of invertible elements of A, respectively. For a subset B of a ring A, the right annihilator and the left annihilator of B in A are denoted by rA(B) and A(B), respectively. We can omit the subscripts if the situation is obvious. If X and Y are subsets of a ring A, then XY denotes the set of all finite sums { ∑ xiyi | xi ∈ X, yi ∈ Y }. In particular, ABA denotes the ideal of A generated by its subset B. A submoduleN of a moduleM is essential (in M) if N has the nonzero intersection with any nonzero submodule of M . In this case, we say that the module M is an essential extension of N . A right (resp. left) module M over a ring A is a nonsingular module if r(m) (resp. (m)) is not an essential right (resp. left) ideal of A for every nonzero element m of M . A ring without nonzero nilpotent elements is called a reduced ring. A ring is normal if all its idempotents are central. Every reduced ring A is normal, since for any idempotent e ∈ A, we have 0 = (eA(1 − e))2 = ((1− e)Ae)2, whence eA(1− e) = (1− e)Ae = 0.

On Rings Whose Primitive Factor Rings Are Left Artinian

In this paper, we prove that if R is a Min-E ring, then the following statements are equivalent: (1) Every left primitive factor ring of R is left artinian; (2) R is a π-regular ring; (3) R is an Exchange ring; (4) R is a Clean ring. As an application, we obtain that if R is Min-E ring, every left primitive factor ring of R is artinian and any direct product of them is a Min-E ring, then every non-zero homomorphic image of R contains only a finite number of prime ideals. When R is Min-E ring and has no infinite set of orthogonal idempotents, a left R-module M is Min-E if and only if M is Max-E. Also, we show, if R is a strongly clean Min-E ring, then R is directly finite and has stable range 1.

Study of Hopficity in Certain Classes of Rings

The main results proved in this paper are: 1. For any non-zero vector space V Dover a division ring D, the ring R= End(V D) is hopfian as a ring 2. Let Rbe a reduced π-regular ring & B(R) the boolean ring of idempotents of R. If B(R) is hopfian so is R.The converse is not true even when Ris strongly regular. 3. Let Xbe a completely regular spaceC(X) (resp. C *(X)) the ring of real valued (resp. bounded real valued) continuous functions on X. Let Rbe any one of C(X) or C *(X). Then Ris an exchange ring if & only if Xis zero dimensional in the sense of Katetov. for any infinite compact totally disconnected space X C(X) is an exchange ring which is not von Neumann regular. 4. Let Rbe a reduced commutative exchange ring. If Ris hopfian so is the polynomial ring R[T 1,…,T n] in ncommuting indeterminates over Rwhere nis any integer ≥ 1. 5. Let Rbe a reduced exchange ring. If Ris hopfian so is the polynomial ring R[T].

Strongly clean rings and fitting's lemma

A ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. These rings are shown to be a natural generalization of the strongly π-regular rings, and several properties of strongly π-regular rings are extended, including their relationship to Fitting's lemma.

Regularity conditions and the simplicity of prime factor rings

R denotes a ring with unity and N r ( R) its nil radical. R is said to satisfy conditions: 1. (1) pm(N r ) if every prime ideal containing N r ( R) is maximal; 2. (2) WCI if whenever a, e ϵ R such that e = e 2, eR + N r ( R) = RaR + N r ( R), and xe − ex ϵ N r ( R) for any x ϵ R, then there exists a positive integer m such that a m (1 − e) ϵ a m N r ( R). For example, if R is right weakly π-regular or every idempotent of R is central, then R satisfies WCI. Many authors have considered the equivalence of condition pm (i.e., every prime ideal is maximal) with various generalizations of von Neumann regularity over certain classes of rings including commutative, PI, right duo, and reduced. In the context of weakly π-regular rings, we prove the following two theorems which unify and extend nontrivially many of the previously known results. Theorem I. Let R be a ring with N r ( R) completely semiprime. Then the following conditions are equivalent: (1) R is right weakly π-regular; (2) R N r(R) is right weakly π-regular and R satisfies WCI; (3) R N r(R) is biregular and R satisfies WCI; (4) for each χ ϵ R there exists a positive integer m such that R = R χ m R + r( χ m ). Theorem II. Let R be a ring such that N r ( R) is completely semiprime and R satisfies WCI. Then the following conditions are equivalent: (1) R is right weakly π-regular; (2) R N r(R) is right weakly π-regular; (3) R N r(R) is biregular; (4) R satisfies pm(N r ); (5) if P is a prime ideal such that N r( R P ) = 0 , then R P is a simple domain; (6) for each prime ideal of R such that N r ( R) ⊆ P, then P = O ̄ P .

Strongly 𝜋-regular rings have stable range one

A ring R R is said to be strongly π \pi -regular if for every a ∈ R a\in R there exist a positive integer n n and b ∈ R b\in R such that a n = a n + 1 b a^{n}=a^{n+1}b . For example, all algebraic algebras over a field are strongly π \pi -regular. We prove that every strongly π \pi -regular ring has stable range one. The stable range one condition is especially interesting because of Evans’ Theorem, which states that a module M M cancels from direct sums whenever End R ( M ) \text {End}_{R} (M) has stable range one. As a consequence of our main result and Evans’ Theorem, modules satisfying Fitting’s Lemma cancel from direct sums.

On strongly pi-regular rings of stable range one

An associative ring R is said to have stable range one if for any a, b ∈ R satisfying aR + bR = R, there exists y ∈ R such that a + by is right (equivalently, left) invertible. Call a ring R strongly π-regular if for every element a ∈ R there exist a number n (depending on a) and an element x ∈ R such that an = an+1x. It is an open question whether all strongly π-regular rings have stable range one. The purpose of this note is to prove the following Theorem: If R is a strongly π-regular ring with the property that all powers of every nilpotent von Neumann regular element are von Neumann regular in R, then R has stable range one.

On N0-quasi-continuous exchange rings

An associative ring R with identity is said to have stable range one if for any a,b∊ R with aR + bR = R, there exists y ∊ R such that a + by is left (equivalently, right) invertible. The main results of this note are Theorem 2: A left or right continuous ring R has stable range one if and only if R is directly finite (i.e xy = 1 implies yx = 1 for all x,y ∊ R), Theorem 6: A left or right N 0o-quasi-continuous exchange ring has stable range one if and only if it is directly finite, and Theorem 12: left or right N 0-quasi-continuous strongly π-regular rings have stable range one. Theorem 6 generalizes a well-known result of Goodearl [10], which says that a directly finite, right N o-continuous von Neumann regular ring is unit-regular

On self-injective strongly π-regular rings

On self-injective strongly π-regular rings

Some characterizations of π-regular rings with no infinite trivial subring

It is shown that a ringRis a π-regular ring with no infinite trivial subring if and only ifRis a subdirect sum of a strongly regular ring and a finite ring. Some other characterizations of such a ring are given. Similar result is proved for a periodic ring. As a corollary, it is shown that every δ-ring is a subdirect sum of a Unite ring and a commutative ring. This was conjectured by Putcha and Yaqub.

Stable range one for rings with many units

The main purpose of this paper is to prove the stable range 1 condition for a number of classes of rings and algebras. Using a modification of a computation of D.V. Tyukavkin, stable range 1 (and a bit more) is obtained from the following simple condition on a ring R: given any x, y ϵ R, there is a unit u ϵ R such that x − u and y − u -1 are both units. Verification of the latter condition then yields stable range 1 in a number of cases, e.g.: (1) any algebra over an uncountable field, in which all non-zero-divisors are units and there are no uncountable direct sums of nonzero one-sided ideals; (2) any algebra over an uncountable field, with only countably many primitive factor rings, all of which are artinian; (3) the endomorphism ring of any noetherian module over an algebra as in (2); (4) any algebraic algebra over an infinite field; (5) any integral algebra over a commutative ring which modulo its Jacobson radical is algebraic over an infinite field; (6) any von Nuemann regular algebra over an uncountable field, which has a rank function. Using other techniques, it is proved that finite Rickart C ∗-algebras, strongly π-regular von Neumann regular rings, and strongly π-regular rings in which every element is a sum of a unit plus a central unit, all have stable range 1. Finally, for an arbitrary commutative ring some overrings with specified stable range properties are constructed, in particular a more or less canonical overring having stable range 1.

On strongly πregular rings and homomorphisms into them

A ring S is strongly π-regular if for every a ∈ S there exists n ≥ 1 such that an∈ an+1S. It is first shown that the dominion and the maximal epimorphic extension of any ring homomorphism α:R →,S, S strongly π-regular, are strongly π-regular. Several results of Schofield on perfect and semiprimary rings are special cases. As an application it is shown that a strongly π-regular ring is a Schur ring, generalizing Lenagan’s theorem for artinian rings. Further if S ⊆ R where S is commutative strongly π-regular then for any finitely generated MSM⊗SR = 0, implies M = 0 , although ⊗Sis not faithful on homomorphisms. Another application shows that the dominion of the natural map Z → S acts like a characteristic for a strongly π-regular ring S A right noetherian ring R satisfying generic regularity, as defined by Goodearl, is shown to have a representation in a strongly π-regular ring [Rcirc] (which is also regular and biregular with simple images artinian) such that Spec [Rcirc] = Spec R , as sets, and Spec [Rcir...

On π-regular rings whose primitive factor rings are artinian

On π-regular rings whose primitive factor rings are artinian

Algebraic algebras over Π-regular rings

We prove here results complementing our work [1b]. In the process of the proofs we extend some classical results from algebraic algebras over fields [5, Theorem 1; 4, Theorem 6.4.3] to algebraic algebras over Π-regular rings (in the sense of Levitski [6]).

Elementary Factorization in π-Regular Rings

This paper extends the results of A. L. Foster (1) on elementary factorization in Boolean-like rings to commutative π-regular rings. After proving some preliminary lemmas we proceed to the partition of the set of non-units of a π-regular ring into irreducible and composite elements. Finally, we prove a number of theorems concerning factorization rings, weakly unique factorization rings, principal ideal rings, etc. The principal result is that a π-regular ring is a weakly unique factorization ring if and only if it is a principal ideal ring.

Rings with Central Idempotent or Nilpotent Elements

It is easy to see (cf. Theorem 1 below) that the centrality of all the nilpotent elements of a given associative ring implies the centrality of every idempotent element; and (Theorem 7) these two properties are in fact equivalent in any regular ring. We establish in this note various conditions, some necessary and some sufficient, for the centrality of nilpotent or idempotent elements in the wider class of π-regular rings (in Theorems 1, 2, 3 and 4 the rings in question are not even required to be π-regular).