论文标题
双kasch戒指
Dual Kasch Rings
论文作者
论文摘要
众所周知，如果每个简单的权利$ r $ $模块都嵌入了投影右$ r $ - 模块中，则环$ r $是正确的kasch。在本文中，我们研究双重概念，并将戒指$ r $右键调用，如果每个简单的权利$ r $ - 模块是一个具有注射式权利$ r $ r $ -MODULE的同构图像。我们证明$ r $是正确的双kasch，并且仅当每个有限生成的投影型正确的$ r $ module都在其注射式船体中锁定。双重kasch环的典型例子是自注射环，V形环和交换完美的戒指。如果该组的命令可逆，有限组的双kasch环偏斜群是双kasch。给出了许多示例，以分离Kasch和Dual Kasch环的概念。结果表明，交换性的kasch环是双kasch，并且只有当它是经典的环时（即每个元素都是零分隔线或可逆的）时，具有有限的Goldie尺寸的交换环为双kasch。对于$ k $，我们获得有限的尺寸$ k $ -algebra是正确的双kasch时，只有当它离开时，它才是正确的。我们还讨论了每个简单的右模块都是其外观船体的同态图像，并且这些环被称为双重双kasch。
It is well known that a ring $R$ is right Kasch if each simple right $R$-module embeds in a projective right $R$-module. In this paper we study the dual notion and call a ring $R$ right dual Kasch if each simple right $R$-module is a homomorphic image of an injective right $R$-module. We prove that $R$ is right dual Kasch if and only if every finitely generated projective right $R$-module is coclosed in its injective hull. Typical examples of dual Kasch rings are self-injective rings, V-rings and commutative perfect rings. Skew group rings of dual Kasch rings by finite groups are dual Kasch if the order of the group is invertible. Many examples are given to separate the notion of Kasch and dual Kasch rings. It is shown that commutative Kasch rings are dual Kasch, and a commutative ring with finite Goldie dimension is dual Kasch if and only if it is a classical ring (i.e. every element is a zero divisor or invertible). We obtain that, for a field $k$, a finite dimensional $k$-algebra is right dual Kasch if and only if it is left Kasch. We also discuss the rings over which every simple right module is a homomorphic image of its injective hull, and these rings are termed strongly dual Kasch.
