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arXiv 0903.2965math.RA
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Noether半局部环上的大投射模

Big projective modules over noetherian semilocal rings

Dolors Herbera, Pavel Prihoda

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41 pages
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摘要：

本文证明了：对于一个恰好具有单右模的$k$同构类的Noether半局部环$R$，可数生成投射右（左）模的同构类的幺半群$V^*(R)$，作为$V^*(R/J(R))$的一个子幺半群，同构于由同余和丢番图线性方程组组成的系统的$(\No \cup\{\infty\})^k$的解幺半群.反之也成立，即如果$M$是$(\No \cup\{\infty\})^k$的一个子幺半群，包含$\No^k$的一个序单位$(n_1,..., n_k)$，$\No^k$是一个同余和线性丢番图方程组的解的集合，那么对于一个Noether半局部环，它可以被实现为$V^*(R)$，使得对于适当的除环$D_1,..., D_k$，$R/J(R)\cong M_{n_1}(D_1)\times ... \times M_{n_k}(D_k)$。

英文摘要：

We prove that for a noetherian semilocal ring $R$ with exactly $k$ isomorphism classes of simple right modules the monoid $V^*(R)$ of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of $V^*(R/J(R))$, is isomorphic to the monoid of solutions in $(\No \cup\{\infty\})^k$ of a system consisting of congruences and diophantine linear equations. The converse also holds, that is, if $M$ is a submonoid of $(\No \cup\{\infty\})^k$ containing an order unit $(n_1,..., n_k)$ of $\No^k$ which is the set of solutions of a system of congruences and linear diophantine equations then it can be realized as $V^*(R)$ for a noetherian semilocal ring such that $R/J(R)\cong M_{n_1}(D_1)\times ... \times M_{n_k}(D_k)$ for suitable division rings $D_1,..., D_k$.

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