Baer rings and quasicontinuous rings have a MDSN

Birkenmeier, Gary F.

doi:10.2140/pjm.1981.97.283

Cited by 40 publications

(18 citation statements)

References 13 publications

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“…The decompositions contain direct summands which are "essentially generated by nilpotents" in the sense that they are essential extensions of (NE(R)), P(R) (i.e., the prime radical of R), or Z2(R) (i.e., the second singular ideal of R). Also, these decompositions are somewhat reminiscent of Kaplansky's theory of types for Baer rings and they extend results in [2] and [22].…”

Section: Introductionsupporting

confidence: 74%

Decompositions of Baer-like rings

Birkenmeier

1992

Acta Math Hung

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Section: Introductionsupporting

confidence: 74%

Decompositions of Baer-like rings

Birkenmeier

1992

Acta Math Hung

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“…We recall that a ring B is reduced if B has no nonzero nilpotent elements, and a ring B is called abelian if every idempotent is central. Reduced rings are abelian and also semiprime (that is, its prime radical is trivial), see [3].…”

Section: -Ringsmentioning

confidence: 99%

“…Birkenmeier in [3], Lemma 1, establishes that if B is a reduced ring, then B is quasi-Baer if and only if B is an abelian Baer ring. This fact together with Proposition 3.5 and Theorem 3.9 guarantee the following result about skew PBW extensions of quasi-Baer rings.…”

Section: Corollary 34mentioning

confidence: 99%

Skew PBW Extensions of Baer, quasi-Baer, p.p. and p.q.-rings

Reyes¹

2015

revint

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“…Then I C (1 -c)R. Let e = 1 -c. By Lemma 1.1, e e S t (R) • Now i(I) n eit = eit n it(l -e) = eit(l -e). [4] Conversely, let / be an ideal of R, and assume there exsits e 6 Si(R) such that / C eR and 1(1) n eR = eR(l -e). Let a € 1 (1).…”

Section: Proposition 12 a Ring R Is Quasi-baer If And Only If Whenementioning

confidence: 99%

“…He then uses this condition to characterise when a finite dimensional algebra with unity over an algebraically closed field is isomorphic to a twisted matrix units semigroup algebra. Further work appeared in [4,6,7,22]. Every prime ring is a quasi-Baer ring.…”

mentioning

confidence: 99%

Quasi-Baer ring extensions and biregrular rings

Birkenmeier¹,

Kim²,

Park³

2000

Bull. Austral. Math. Soc.

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A ring R with unity is called a (quasi-) Baer ring if the left annihilator of every (left ideal) nonempty subset of R is generated (as a left ideal) by an idempotent. Armendariz has shown that if R is a reduced Pi-ring whose centre is Baer, then R is Baer. We generalise his result by considering the broader question: when does the (quasi-) Baer condition extend to a ring from a subring? Also it is well known that a regular ring is Baer if and only if its lattice of principal right ideals is complete. Analogously, we prove that a biregular ring is quasi-Baer if and only if its lattice of principal ideals is complete.Throughout R will denote a ring with unity, B ( # ) its set of central idempotents, Cen (R) its centre, Z T {R) and Zi(R) its right and left singular ideals, respectively. All subrings have a unity which may not coincide with the unity of the overring. The word "ideal" used alone (that is, without the adjectives "left" or "right") means a two-sided ideal. Recall from [19] that R is a Baer ring if the right annihilator of every nonempty subset of R is generated (as a right ideal) by an idempotent. The study of Baer rings has its roots in functional analysis [3] and [19]. In [19] Kaplansky introduced Baer rings to abstract various properties of von Neumann algebras and complete regular *-rings. The class of Baer rings includes the von Neumann algebras (for example, the algebra of all bounded operators on a Hilbert space), the commutative C* -algebras C(T) of continuous complex valued functions on a Stonian space T, and the regular rings whose lattice of principal right ideals is complete (for example, regular rings which are right continuous or right self-injective). Also the Sherman-Takeda theorem [24] and [28] shows that every C* -algebra has a universal enveloping von Neumann algebra (hence a Baer ring).In [12] Clark defines a ring to be quasi-Baer if the left annihilator of every ideal is generated (as a left ideal) by an idempotent (equivalently, the left annihilator of every

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Baer rings and quasicontinuous rings have a MDSN

Cited by 40 publications

References 13 publications

Decompositions of Baer-like rings

Decompositions of Baer-like rings

Skew PBW Extensions of Baer, quasi-Baer, p.p. and p.q.-rings

Quasi-Baer ring extensions and biregrular rings

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