论文标题
MacWilliams延伸条件和准杂种环
MacWilliams extending conditions and quasi-Frobenius rings
论文作者
论文摘要
麦克维利亚姆斯（MacWilliams）证明，每个有限的场均具有锤子重量的扩展特性，后来在伍德（Wood）的开创性工作中扩展了锤子，这些伍德（Wood）将有限的Frobenius Rings描述为满足MacWilliams扩展特性的那些戒指。在本文中，何时解决了麦克维利亚姆戒指的问题。事实证明，右或左noetherian左1-Macwilliams戒指是Quasi-frobenius，因此回答了[M. C. Iovanov，关于无限的Macwilliams环和最小的注射率条件，Proc。阿米尔。数学。 Soc。，doi：10.1090/proc/15929]和[F. M. Schneider，J。Zumbrägel，MacWilliams的“无限戒指的扩展定理”，Proc。阿米尔。数学。 Soc。 147，3（2019），947-961]。我们还证明，正确的完美，左构态不变的戒指是自注明的。特别是，如果$ r $是正确的（或左）的Artinian，则左图为止，$ r $是Quasi-frobenius，因此回答了[M. C. Iovanov，关于无限的Macwilliams环和最小的注射率条件，Proc。阿米尔。数学。 Soc。，doi：10.1090/proc/15929]。
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 [M. C. Iovanov, On infinite MacWilliams rings and minimal injectivity conditions, Proc. Amer. Math. Soc., DOI: 10.1090/proc/15929] and [F. M. Schneider, J. Zumbrägel, MacWilliams' extension theorem for infinite rings, Proc. Amer. Math. Soc. 147, 3 (2019), 947-961]. We also prove that a right perfect, left automorphism-invariant ring is left self-injective. In particular, this yields that if $R$ is a right (or left) artinian, left automorphism-invariant ring, then $R$ is quasi-Frobenius, thus answering a question asked in [M. C. Iovanov, On infinite MacWilliams rings and minimal injectivity conditions, Proc. Amer. Math. Soc., DOI: 10.1090/proc/15929].
