The preservation of the semiprime Goldie property by strong semilattice sums

Ho, Yue-Chan Phoebe

doi:10.1090/s0002-9939-1992-1074752-x

Cited by 3 publications

(2 citation statements)

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“…A cornmutative band is called a semilattice. Semilattice-graded rings were considered in [2], [7], [ll], [22] and other papers. Then R = es,, R, is an S-graded ring, and Re = 0 for all idempotents e E S .…”

Section: S-graded Ring Intersects a Jnite Number Of Maximal Subgroupsmentioning

confidence: 99%

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Semisimple artinian graded rings

Cazaran¹,

Kelarev²

1999

Communications in Algebra

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Section: S-graded Ring Intersects a Jnite Number Of Maximal Subgroupsmentioning

confidence: 99%

“…If B is a semilattice, then all special B-graded rings are strong semilattice sums of rings (see [2] for definition). Proof is similar to the proof of the main theorem of [7], and so it will be omitted. 0 A Brandt semigroup is an inverse completely 0-simple semigroup.…”

Section: Downloaded By [Byu Brighammentioning

confidence: 99%

Semisimple artinian graded rings

Cazaran¹,

Kelarev²

1999

Communications in Algebra

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Distributive semigroup rings and related topics

Tuganbaev¹

1999

J Math Sci

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A. A. Tuganbaev UDC 512.55 All rings are assumed to be associative and with nonzero identity element. Expressions such as % Noetherian ring" mean that the corresponding right and left conditions hold.We denote by End(M), J(M), and Lat(M) the endomorphism ring, the Jacobson radical, and the lattice of all submodules of a module M, respectively.A module M is called distributive if the lattice Lat(M) of all submodules of M is distributive, i.e., FO(G+H) = F71G+FOH for allsubmodules F, G, and H of the module M. A moduleMis called uniserial if any of its submodules are comparable with respect to inclusion, i.e., the lattice Lat(M) is a chain.If A is a commutative regular ring (e.g., a field) and G is a locally cyclic group, then the group ring A[G] and the polynomial rings A[x], A[x, x -1] are distributive Bezout rings.All strongly regular rings (e.g., all factor rings of direct products of skew fields and all commutative regular rings) are distributive; all commutative Dedekind rings (e.g., rings of integral algebraic numbers or commutative principal ideal rings) and all valuation rings in division rings are distributive.Distributive group and semigroup rings were studied in [8,20,17,41,43,44,46,47,49]. Distributive quaternion algebras were studied in [50][51][52]. In [42,44,48], some applications of distributive rings and modules to formal power series rings were obtained. Distributive graded modules were considered in [6]. Distributive quaternion algebras were studied in [50][51][52]. Distributive modules over incidence algebras were studied in [14]. In [64], distributive rings are used for the study of rings of continuous functions in topological spaces. The results concerning distributive modules and rings were addressed in surveys [4,23,24]. If B is a subset of a ring A, then Id(B), r(B), and s are the ideal of A generated by B, the right annihilator of B, and the left annihilator of B, respectively. The group of invertible elements of a ring A is denoted by U(A).A ring is normal if all its idempotents are central. A ring A with no nonzero nilpotent elements is called a reduced ring.A ring A is called regular if for any f E A, there exists g E A such that f = fgf. A ring A is called strongly regular if the following equivalent conditions hold. (i) Given any a E A, there exists b E A such that a = a~b.(ii) Given any a E A, there exists b E A such that a = ba 2.(iii) Every element of A is a product of a central idempotent and a unit. (iv) A is a regular reduced ring.(v) A is a regular normal ring.A muItiplicative set in a ring A is any subset T of A such that 1 E T, 0 q~ T, and T is closed under multiplication.A completely prime ideal in a ring A is any proper ideal B such that A \ B is multiplicative set (i.e., if A/B is a domain).A subset T in a ring A is right permutable if for any a E A and t E T, there exist b E A and u E T such that au = tb.A subset T in a ring A is called right reversible (weakly right reversible) if given any a E A and t E T such that ta = 0 (such that a 2 = 0 and ta = 0), there exists...

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