Let R be a unit-regular ring, and let a, b, c ? satisfy aba = aca. If ac or ba is Drazin invertible, we prove that their inverses are similar. Furthermore, if group then similar to ba. For any n?n complex matrices A, B,C with ABA ACA, AC BA only k-powers have the same rank. These generalize known Flanders? theorem proved by Hartwig.

Let $X$ be a set and $\mathcal{T}_X$ the full transformation semigroup on $X$. For partition $\mathcal{P}$ of $X$, we consider semigroups $T(X, \mathcal{P}) = \{f\in \mathcal{T}_X| (\forall X_i\in (\exists X_j \in \mathcal{P})\;X_i f \subseteq X_j\}$, $\Sigma(X, T(X, \mathcal{P})|(\forall X_i \mathcal{P})\; Xf \cap \neq \emptyset\}$, $\Gamma(X, \mathcal{T}_X|(\forall \mathcal{P})(\exists X_j\in...

We presented a simple and direct way to construct unipotent unit clean but not nil-clean element in ring. New examples of unipotent/unit-regular elements that are given. also study the product two idempotents/unit-regulars which unit-regular. The studies exemplified subrings M_2 (Z).

Resolution refinements called w-resolution trees with lemmas (WRTL) and with input lemmas (WRTI) are introduced. Dag-like resolution is equivalent to both WRTL and WRTI when there is no regularity condition. For regular proofs, an exponential separation between regular dag-like resolution and both regular WRTL and regular WRTI is given. It is proved that DLL proof search algorithms that use cla...