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Goldie’s theorems for involution rings. (English) Zbl 0810.16034

In the consistent theory of rings with involution only symmetric notions may occur (e.g. no one-sided ideals). The author gives the involutive versions of Goldie’s Theorems. If \(R\) is a ring with involution, then a right ring of quotients \(Q\) is also a left ring of quotients, and the involution of \(R\) extends uniquely to \(Q\). A subring \(B\) of \(R\) is called a biideal, if \(BRB \subseteq B\). The symmetric versions of Goldie’s Theorems say that a (semi)prime ring \(R\) (regardless of involution) is Goldie if and only if \(R\) satisfies the a.c.c. on annihilator biideals and the maximum condition on biideal direct sums. A biideal \(B\) of a ring \(R\) with involution is called a *-biideal, if \(B^* = B\). Using this notion, Goldie involution rings are defined. An involution ring \(R\) is a Goldie ring if and only if it is a Goldie involution ring. The involutive versions of Goldie’s Theorems are explicitly given; the point is that the *-ring of quotients \(Q\) of a *-prime involution ring \(R\) may be either a matrix ring or the direct sum of a matrix ring and its opposite ring endowed with the exchange involution.

MSC:

16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16N60 Prime and semiprime associative rings
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References:

[1] Amitsur S. A., GK-dimension of corners and ideals 69 pp 152– (1990) · Zbl 0697.16025
[2] Beidar K.I., Rings with involution and Chain conditions, Preprint · Zbl 0787.16021
[3] Loi N. V., On the structure of semiprirne involution rings 7 pp 153– (1988)
[4] Steinfeld O., Quasi ideals in rings and semigroups (1978) · Zbl 0403.16001
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