Generalized inverses: Uniqueness proofs and three new classes
Generalized inverses: Uniqueness proofs and three new classes
复制标题
DOI：
10.1016/j.laa.2014.02.034
复制
发表时间：
2014-05
影响因子：
1.1
通讯作者：
M. Drazin
中科院分区：
文献类型：
--
作者：
M. Drazin
给定任意含1的环R和任意a，B，c∈ R，则推广了JJ Koliha和P. Patricio在2002年和Z. Wang和J. Chen在2012年提出的一个新的概念，a称为“（B，c）-伪极”，如果存在幂等元p∈ R使得1− p∈（B R+ J）（Rc + J），p B和cp ∈ J（其中J表示R的Jacobson根），p位于a的第二交换子中。这个p被证明是唯一的，只要它存在。定义了a的一个新的外广义逆y，称为a的（B，c）-伪逆，证明了y的存在性意味着a是（B，c）-伪极的，因此y本身是唯一的.推广Koliha，Patrício，Wang和Chen的结果，进一步发现了（B，c）-伪极性质和（B，c）-伪可逆性质之间的联系，并证明了a1 a 2的（B，c）-伪可逆性暗示了a2 a 1的相应性质.另外还引入了两种类型的广义定义的外广义逆。
Given any ring R with 1 and any a, b, c∈ R, then, generalizing ideas of JJ Koliha and P. Patrício in 2002 and of Z. Wang and J. Chen in 2012, a is called “(b, c)-pseudo-polar” if there exists an idempotent p∈ R such that 1− p∈(b R+ J)∩(R c+ J), p b and c p∈ J (where J denotes the Jacobson radical of R) and p lies in the second commutant of a. This p is shown to be unique whenever it exists. A new outer generalized inverse y of a, called the (b, c)-pseudo-inverse of a, is also defined, and the existence of y is shown to imply that a is (b, c)-pseudo-polar, and hence that y is itself unique. Generalizing results of Koliha, Patrício, Wang and Chen, further connections between the (b, c)-pseudo-polar and (b, c)-pseudo-invertible properties are found, and the (b, c)-pseudo-invertibility of a 1 a 2 is shown to imply a corresponding property for a 2 a 1. Two further types of uniquely-defined outer generalized inverses are also introduced.