Projective modules over semilocal rings. (English) Zbl 0981.16003

Huynh, D. V. (ed.) et al., Algebra and its applications. Proceedings of the international conference, Athens, OH, USA, March 25-28, 1999. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 259, 181-198 (2000).
In the paper semigroup means an additive, commutative semigroup with zero, and ring means an associative ring with identity \(1\neq 0\). If \(R\) is a ring, by \(J(R)\) is denoted the Jacobson radical of \(R\).
If \(R\) is a semilocal ring, the canonical projection \(\pi\) of \(R\) onto the semisimple ring \(R/J(R)\) induces a semigroup homomorphism \(S_\oplus(\pi)\colon S_\oplus(P\text{-Mod }R)\to S_\oplus(P\text{-Mod }R/J(R))\), where \(_\oplus(P\text{-Mod }R)\) denotes the semigroup of isomorphism classes of finitely generated projective modules over the ring \(R\); this mapping \(S_\oplus(\pi)\) turns out to be a full affine embedding of the semigroup \(S_\oplus(P\text{-Mod }R)\) into the free commutative semigroup \(S_\oplus(P\text{-Mod }R/J(R))\).
In a previous paper the authors proved that every full affine embedding of a semigroup \(M\) into a free commutative semigroup \(\mathbb{N}^n\) can be realized as the mapping \(S_\oplus(\pi)\colon S_\oplus(P\text{-Mod }R)\to S_\oplus(P\text{-Mod }R/J(R))\) for a suitable (hereditary) semilocal ring \(R\).
The aim of this paper is to provide the reader with the necessary framework and help him to appreciate the significance of the above result. The authors recall some elementary properties of full semigroups and full affine semigroups and they give examples and applications of these properties to the semigroup \(S_\oplus(P\text{-Mod }R)\) of a semilocal ring \(R\). Also, the authors consider the embeddability of semilocal semihereditary rings into semisimple Artinian rings, simple Artinian rings and division rings.
For the entire collection see [Zbl 0947.00022].


16D40 Free, projective, and flat modules and ideals in associative algebras
16L30 Noncommutative local and semilocal rings, perfect rings
16E20 Grothendieck groups, \(K\)-theory, etc.