Generalizations of some theorems in group and ring theory to skewrings

Abstract

A triple (R, +, .) is called a skewring if and only if 1) (R, +) is a group, 2) (R, .) is a semigroup and 3) for any x, y, z R, x(y+z) = xy+xz and (x+y)z = xz+yz. Let I be a nonempty subset of a skewring R. Then I is called a subskewring of R if and only if I is a skewring under the operations of R and I is called a normal ideal of R if and only if I is a subskewring of R and for any r R, x I, rx, xr, r+x-r I. Let p be an equivalence relation on a skewring R. Then p is called a congruence on R if and only if for any x, y, z R, xpy implies (x+z)p(y+z), (z+x)p(z+y), (xz)p(yz) and (zx)p(zy). Let L(R) be the set of all congruences on a skewring R. For any p, sigma L(R), define p