Semilocal Rings and Related Rings

The purpose of this lectures is giving an introduction to the theory of non-commutative semilocal rings (A ring R is said to be semilocal if R/J(R) is semisimple Artinian). A class of semilocal rings include all division rings, left and right Artinian rings, perfect rings, semi-perfect rings, semiprimary rings, local rings, and homogeneous semilocal rings. If R is a ring, S is a subring of R, and T is an extension of R the main question arise what is the relations hold between these rings?. In general, we certainly cannot expect that a property of any one of these rings transfer directly to other rings. In this short course we shall recognize various characterizations of subrings and an extensions of semilocal rings. Also, we shall investigate the necessary and sufficient conditions which grantee transferring some particular algebraic properties between R, S, and T.