Abstract: Let M be a monoid andα an endomorphism of a ring R. We investigate the properties of M-symmetric rings with endomorphisms on R by introducing the concept of strongly M-α-symmetric ring. Using the classical method in ring theory, we prove that(1)R is a strongly right M-α-symmetric ring if and only if R is a strongly left M-α-symmetric ring;(2)Letαbe an automorphism of R. If R is a right Ore ring, hen R is a strongly M-α-symmetric ring if and only if Q is a strongly M-a-symmetric ring, where Q is the classical right quotient ring of the R.

Key words: Symmetric ring, Stronglyα-symmetric ring, Strongly M-symmetric ring, Strongly M-α-symmetric ring