Rings of quotients of rings without nilpotent elements

Steinberg, Stuart A.

doi:10.2140/pjm.1973.49.493

Cited by 11 publications

(6 citation statements)

References 18 publications

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“…Rings which have the property that prime factor rings are domains include commutative rings as well as rings in which prime factor rings are division rings. Hence we immediately have the following corollary which was discovered independently by various authors including Herstein, Snider [27], Steinberg [28], and Wong [30]. A prime ideal P of R is completely prime if R/P is a domain.…”

Section: A Ring R Is Von Neumann Regular If and Only If Each Factor Rmentioning

confidence: 72%

On the von Neumann regularity of rings with regular prime factor rings

Fisher¹,

Snider²

1974

Pacific J. Math.

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Section: A Ring R Is Von Neumann Regular If and Only If Each Factor Rmentioning

confidence: 72%

On the von Neumann regularity of rings with regular prime factor rings

Fisher¹,

Snider²

1974

Pacific J. Math.

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“…Since G is reduced the singular ideals of G are zero. Thus [6] H is a quotient ring of G and so H is reduced [15].…”

Section: Lemma 34 // G Is Reduced and Large In H Then H Is Reducedmentioning

confidence: 99%

The hulls of semiprime rings

Conrad¹

1978

Czech. Math. J.

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“…Then obviously, R C Q(R) It is proved by Wong and Johnson [14] that Q(R) is a subring of Q r (R) and it is unique (up to isomorphism over R) maximal two sided ring of quotients of R. Also for every reduced ring i?, Q(R) is reduced (see Steinberg [12], page 466). It is proved in [10] (page 483) that every orthogonal subset X of R has a supremum in Q(R) and since Q(Q(R)) = Q(R), Q(R) is orthogonally complete.…”

Section: R K Raimentioning

confidence: 99%

On orthogonal completion of reduced rings

M.K.¹

1983

Pacific J. Math.

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It was proved by the author earlier that every orthogonal extension of a reduced ring R is a subring of Q(R), the maximal two sided ring of quotients oiR and the orthogonal completion of R, if it exists, is unique upto an isomorphism. Here, in Theorem 2, we prove that the orthogonal completion of R, if it exists, is a ring of right quotients Q F (R) of R with respect to an idempotent filter F of dense right ideals of R. Introduction. Abian [2] showed that the canonical order relation ' < ' of Boolean rings can be defined for reduced rings R (a ring with no nonzero nilpotent element) by writing a < b if ab -a 2 and this order relation makes R into a partially ordered multiplicative semigroup. Reduced rings under this relation ' < ' were studied by Abian [1] and Chacron [5] to characterise the direct produce of integral domains, division rings and fields. Their studies involved the concepts of orthogonal completeness and orthogonal completion of reduced rings. These two concepts, on their own merit, were studied by Burgess, Raphael and Stephenson [3], [4], [11]. They proved that reduced rings which have enough idempotents (/-dense) or satisfy certain chain conditions have an orthogonal completion. In this paper we shall provide a necessary and sufficient condition for a reduced ring to have an orthogonal completion.

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Rings of quotients of rings without nilpotent elements

Cited by 11 publications

References 18 publications

On the von Neumann regularity of rings with regular prime factor rings

On the von Neumann regularity of rings with regular prime factor rings

The hulls of semiprime rings

On orthogonal completion of reduced rings

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