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Browsing NIR by Subject "(bc)-inverse"

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    Equivalence Relations Based on (b,c)-Inverses in Rings
    (Springer Nature, 2024) Stanišev, Ivana; Višnjić, Jelena; Djordjevic, Dragan
    In this paper, we introduce a binary relation in the set of all (b, c)-invertible elements of a ring, defined in the manner like minus, star, sharp, core, and dual core partial orders. We prove that this binary relation is actually an equivalence relation and we investigate some of its properties. Furthermore, we define another equivalence relation on the set of all (b, c)-invertible elements of a ring, as an improvement of the previous mentioned equivalence relation. In addition, the equivalence criteria are given for the equality of (b, c)-related idempotents of two elements and their (b, c)-invertibility is studied.
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    Reverse and Forward Order Law for the (b,c)-inverse
    (Univerzitet u Beogradu, Matematički fakultet, 2024) Višnjić, Jelena; Stanišev, Ivana; Ke, Yuanyuan
    In a ring R it is well known that if a, w ∈ R^(-1), then aw is also invertible and (aw)^−1 =w^−1 a^−1. This property is called the reverse order law. As the reverse order law holds for the classical inverse, the topic of necessary and sufficient conditions for generalized inverses became very often in the past decades. The similar property, (aw)^−1 = a^−1 w^−1, is known as the forward order law. Contrary to the reverse order law, even if a and w are both invertible, the forward order law is not valid in general. Here we present results for the reverse order law for the (b, c)-inverse in a unital ring. An equivalent condition for this law to hold for the (b, c)-inverse is derived. Furthermore, the forward order law for the (b, c)-inverse in a ring with a unity is introduced for different choices of b and c. Moreover, as corollaries of obtained results, equivalent conditions for the reverse order law and the forward order law for the inverse along an element are derived.
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    Some properties of (b, c)-inverses in rings
    (Taylor & Francis, 2023) Višnjić, Jelena; Stanišev, Ivana; Ke, Yuanyuan
    In this paper, we study (b, c)-invertibility of some arbitrary (b, c)-inverse and we give an equivalent condition for its (b, c)-invertibility. Moreover, we investigate when some well known properties of generalized inverses hold for (b, c)-inverses. Further, we study when the (b, c)-inverse of the unity is equal to the unity itself and we obtain equivalent conditions for this to hold. In addition, we investigate the reverse order law for the (b, c)-inverse. Finally, we prove that in the case when the unity is (b, c)-invertible, the set of all (b, c)-inverses, with respect to multiplication, is a group.