MacWilliams extending conditions and quasi-Frobenius rings
MacWilliams extending conditions and quasi-Frobenius rings
复制标题
麦克威廉斯扩展条件和准弗罗贝尼乌斯环
DOI：
10.1016/j.jalgebra.2022.05.005
复制
发表时间：
2022
影响因子：
0.9
通讯作者：
A. K. Srivastava
中科院分区：
文献类型：
--
作者：
P. A. Guil Asensio;A. K. Srivastava
MacWilliams证明了每一个有限域都具有Hamming权的可拓性，后来Wood在其开创性的工作中对其进行了推广，Wood将有限Frobenius环描述为满足MacWilliams可拓性的环。本文讨论了MacWilliams环何时为拟frobenius环的问题。证明了一个右或左noetherian左1-MacWilliams环是拟frobenius，从而回答了[13]，[22]中提出的不同问题。我们还证明了一个右完备的左自同构不变环是左自内射的。特别地，这产生了ifRis是一个右（或左）环，左自同构不变环，那么ris是一个拟frobenius环，从而回答了[13]中提出的问题。
MacWilliams proved that every finite field has the extension property for Hamming weight which was later extended in a seminal work by Wood who characterized finite Frobenius rings as precisely those rings which satisfy the MacWilliams extension property. In this paper, the question of when is a MacWilliams ring quasi-Frobenius is addressed. It is proved that a right or left noetherian left 1-MacWilliams ring is quasi-Frobenius thus answering the different questions asked in [13], [22]. We also prove that a right perfect, left automorphism-invariant ring is left self-injective. In particular, this yields that ifRis a right (or left) artinian, left automorphism-invariant ring, thenRis quasi-Frobenius, thus answering a question asked in [13].