S-Cohn–Jordan Extensions

Abstract

Let a monoid S act on a ring R by injective endomorphisms and A(R; S) denote the S-Cohn–Jordan extension of R. A series of results relating properties of R and that of A(R; S) are presented. In particular it is shown that: (1) A(R; S) is semiprime (prime) iff R is semiprime (prime), provided R is left Noetherian; (2) if R is a semiprime left Goldie ring, then so is A(R; S), Q(A(R; S)) = A(Q(R); S) and udim R = udim A; (3) A(R; S) is semisimple iff R is semisimple, provided R is left Artinian. Some applications to the skew semigroup ring R#S are given.

Keywords:

• Left Goldie rings
• S-Cohn–Jordan extensions
• Semiprime rings
• Uniform dimension

Mathematics Subject Classification:

• Primary 16S60, 16W20
• Secondary 16P60, 16P40

ACKNOWLEDGMENTS

I wish to express my thanks to Andre Leroy both for helpful conversations and the kind interest in the progress of this work. In particular, it was Andre who pointed me out that one can find a construction of A(R; S) in the book Cohn (1965).

The research was supported by Polish KBN grant No. 1 P03A 032 27.

Notes

Communicated by E. R. Puczyłowski.