TY - JOUR

T1 - On classical quotients of polynomial identity rings with involution

AU - Rowen, Louis Halle

PY - 1973/9

Y1 - 1973/9

N2 - Let (R, *) denote a ring R with involution (*), where “involution„ means “anti-automorphism of order ≦ two„. We can specialize many ring-theoretical concepts to rings with involution; in particular an ideal of (R, *) is an ideal of R stable under (*), and the center of (R, *) is the set of central elements of R which are fixed under (*). Then we say (R, *) is prime when the product of any two nonzero ideals of (R, *) is nonzero; similarly (R, *) is semiprime when any power of a nonzero ideal of (R, *) is nonzero. The main result of this paper is a strong analogue to Posner’s theorem [5], namely that any prime (R, *) with polynomial identity has a ring of quotients RT, formed merely by adjoining inverses of nonzero elements of the center of (R, *). This quotient ring (RT, *) is simple and finite dimensional over its center. An extension of these results to semiprime Goldie rings with polynomial identity is given.

AB - Let (R, *) denote a ring R with involution (*), where “involution„ means “anti-automorphism of order ≦ two„. We can specialize many ring-theoretical concepts to rings with involution; in particular an ideal of (R, *) is an ideal of R stable under (*), and the center of (R, *) is the set of central elements of R which are fixed under (*). Then we say (R, *) is prime when the product of any two nonzero ideals of (R, *) is nonzero; similarly (R, *) is semiprime when any power of a nonzero ideal of (R, *) is nonzero. The main result of this paper is a strong analogue to Posner’s theorem [5], namely that any prime (R, *) with polynomial identity has a ring of quotients RT, formed merely by adjoining inverses of nonzero elements of the center of (R, *). This quotient ring (RT, *) is simple and finite dimensional over its center. An extension of these results to semiprime Goldie rings with polynomial identity is given.

KW - Center

KW - Central quotients

KW - Classical ring of quotients

KW - Goldie ring

KW - Involution

KW - Polynomial identity

KW - Prime

KW - Semiprime

KW - Simple

UR - http://www.scopus.com/inward/record.url?scp=84911698835&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-1973-0323822-8

DO - 10.1090/S0002-9939-1973-0323822-8

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AN - SCOPUS:84911698835

SN - 0002-9939

VL - 40

SP - 23

EP - 29

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 1

ER -