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Keywords:
exchange ring; ideal; related comparability
exchange ring; ideal; related comparability
Summary:
In this paper, we introduce related comparability for exchange ideals. Let $I$ be an exchange ideal of a ring $R$. If $I$ satisfies related comparability, then for any regular matrix $A\in M_n(I)$, there exist left invertible $U_1,U_2\in M_n(R)$ and right invertible $V_1,V_2\in M_n(R)$ such that $U_1V_1AU_2V_2= \operatorname{diag}(e_1,\cdots ,e_n)$ for idempotents $e_1,\cdots ,e_n\in I$.
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