Reverse Order Law for the Core Inverse in Rings

Reverse Order Law for the Core Inverse in Rings In this paper, necessary and sufficient conditions of the one-sided reverse order law $$(ab)^{\tiny {\textcircled {\tiny \#}}}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}$$ ( a b ) # = b # a # , the two-sided reverse order law $$(ab)^{\tiny {\textcircled {\tiny \#}}}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}$$ ( a b ) # = b # a # and $$(ba)^{\tiny {\textcircled {\tiny \#}}}=a^{\tiny {\textcircled {\tiny \#}}}b^{\tiny {\textcircled {\tiny \#}}}$$ ( b a ) # = a # b # for the core inverse are given in rings with involution. In addition, the mixed-type reverse order laws, such as $$(ab)^{\#}=b^{\tiny {\textcircled {\tiny \#}}}(abb^{\tiny {\textcircled {\tiny \#}}})^{\tiny {\textcircled {\tiny \#}}}$$ ( a b ) # = b # ( a b b # ) # , $$a^{\tiny {\textcircled {\tiny \#}}}=b(ab)^{\#}$$ a # = b ( a b ) # and $$(ab)^{\#}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}$$ ( a b ) # = b # a # , are also considered. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

Reverse Order Law for the Core Inverse in Rings

17 pages

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Abstract

In this paper, necessary and sufficient conditions of the one-sided reverse order law $$(ab)^{\tiny {\textcircled {\tiny \#}}}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}$$ ( a b ) # = b # a # , the two-sided reverse order law $$(ab)^{\tiny {\textcircled {\tiny \#}}}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}$$ ( a b ) # = b # a # and $$(ba)^{\tiny {\textcircled {\tiny \#}}}=a^{\tiny {\textcircled {\tiny \#}}}b^{\tiny {\textcircled {\tiny \#}}}$$ ( b a ) # = a # b # for the core inverse are given in rings with involution. In addition, the mixed-type reverse order laws, such as $$(ab)^{\#}=b^{\tiny {\textcircled {\tiny \#}}}(abb^{\tiny {\textcircled {\tiny \#}}})^{\tiny {\textcircled {\tiny \#}}}$$ ( a b ) # = b # ( a b b # ) # , $$a^{\tiny {\textcircled {\tiny \#}}}=b(ab)^{\#}$$ a # = b ( a b ) # and $$(ab)^{\#}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}$$ ( a b ) # = b # a # , are also considered.

Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: Jun 5, 2018

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