态射和的核与双核逆
Core and Dual Core Inverses of a Sum of Morphisms
摘要：
设$\mathscr{C}$是一个具有对合$\ast$的加法范畴.假设$φ: X \rightarrow X$是具有核心逆$φ^{\co} : X \rightarrow X$的$\mathscr{C}$的态射，并且$η: X \rightarrow X$是$\mathscr{C}$的态射使得$1_X+φ^{\co}η$是可逆的。令$α=(1_X+φ^{\co}η)^{-1},$ $β=(1_X+ηφ^{\co})^{-1},$ $\varepsilon=(1_X-φφ^{\co})ηα(1_X-φ^{\co}φ),$ $γ=α(1_X-φ^{\co}φ)β^{-1}φφ^{\co}β,$ $σ=αφ^{\co}φα^{-1}(1_X-φφ^{\co})β,$ $δ=β^{\ast}(φ^{\co})^{\ast}η^{\ast}(1_X-φφ^{\co})β.$则$f=φ+η-\varepsilon$有核逆当且仅当$1_X-γ$，$1_X-σ$和$1_X-δ$可逆.此外，给出了$f$的核逆的表达式。设$R$是一个有单位元的$\ast$ -环，$J(R)$是它的Jacobson根，若$a\in R^{\co}$具有核逆$a^{\co}$和$j\in J(R)$，则$a+j\in R^{\co}$当且仅当$(1-aa^{\co})j(1+a^{\co}j)^{-1}(1-a^{\co}a)=0$ .对于对偶核逆也给出了类似的结果。
英文摘要：
Let $\mathscr{C}$ be an additive category with an involution $\ast$. Suppose that $φ: X \rightarrow X$ is a morphism of $\mathscr{C}$ with core inverse $φ^{\co} : X \rightarrow X$ and $η: X \rightarrow X$ is a morphism of $\mathscr{C}$ such that $1_X+φ^{\co}η$ is invertible. Let $α=(1_X+φ^{\co}η)^{-1},$ $β=(1_X+ηφ^{\co})^{-1},$ $\varepsilon=(1_X-φφ^{\co})ηα(1_X-φ^{\co}φ),$ $γ=α(1_X-φ^{\co}φ)β^{-1}φφ^{\co}β,$ $σ=αφ^{\co}φα^{-1}(1_X-φφ^{\co})β,$ $δ=β^{\ast}(φ^{\co})^{\ast}η^{\ast}(1_X-φφ^{\co})β.$ Then $f=φ+η-\varepsilon$ has a core inverse if and only if $1_X-γ$, $1_X-σ$ and $1_X-δ$ are invertible. Moreover, the expression of the core inverse of $f$ is presented. Let $R$ be a unital $\ast$-ring and $J(R)$ its Jacobson radical, if $a\in R^{\co}$ with core inverse $a^{\co}$ and $j\in J(R)$, then $a+j\in R^{\co}$ if and only if $(1-aa^{\co})j(1+a^{\co}j)^{-1}(1-a^{\co}a)=0$. We also give the similar results for the dual core inverse.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
