In 2020, Ferreyra et al. defined the weak core inverse of a complex matrix. We generalize it to a unitary ring with involution and investigate the relationship between the weak core inverse and the m-weak group inverse. Furthermore, we prove that an element is pseudo core invertible if this element is both {1, 3}-invertible and weak group invertible. Then, we show that each nilpotent element is Moore–Penrose invertible if and only if each pseudo core invertible element is weak core invertible. Finally, we consider the conditions under which the weak core inverse of an element coincides with another generalized inverse.

中文翻译：

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对合环中的弱核反演和伪核反演

2020 年，Ferreyra 等人。定义了复数矩阵的弱核逆。我们将其推广到具有对合的酉环，并研究弱核逆与m -弱群逆之间的关系。此外，如果该元素既是{1, 3}-可逆又是弱群可逆，我们证明该元素是伪核心可逆。然后，我们证明每个幂零元都是 Moore–Penrose 可逆当且仅当每个伪核心可逆元素都是弱核心可逆时。最后，我们考虑一个元素的弱核心逆与另一个广义逆重合的条件。