The weak Moore-Penrose inverse of elements in rings with involution

Let R be a ring with involution. In this paper, we introduce a new type of generalized inverse called weak Moore-Penrose inverse in R. The weak Moore- Penrose inverse generalizes the notion of weak dual generalized inverse, introduced by Li and Wang for dual matrices in 2023, to the case of ∗-ring R. An element a ∈ R is weak Moore-Penrose invertible if there exists some x ∈ R such that a∗ axaa∗ = a∗ aa∗, xax = x, (ax)∗ = ax and (xa)∗= xa. If such an x exists, then it is unique (denoted by aw†). We prove that aw† exists if and only if (a∗aa∗)† exists, in which case aw† = a∗(a∗aa∗)†a∗. Then, we consider the relationship between weak Moore- Penrose inverse and other generalized inverses. Finally, the equivalent conditions under which aw† is normal, Hermitian and idempotent are investigated, respectively.