子环扩张的morphic性质

设R是一个环,C是R的子环,C包含环R的单位元.令CR={(c,r)|c∈C,r∈R},按方式(c1,r1)+(c2,r2)=(c1+c2,r1+r2)和(c1,r1)·(c2,r2)=(c1c2,c1r2+r1c2+r1r2)定义加法和乘法,易证CR是环,且单位元为(1R,0),故称这样的环为R的子环扩张.特别的,当子环C就取环R本身时,称R×R为R的平凡子环扩张.文章给出一些相关性质和例子,并证明了:1)若S=C×R是morphic环,则C和R也都是morphic环;2)若R是半单环,则R的平凡子环扩张是强morphic环.

The Morphic Properties of Subring-Extension

Let R be a ring, C be a subring of R, and 1R∈C. Set C(∝)R={(c,r)|c∈C,r∈R},with the addition and multiplication defined (c1,r1)+(c2,r2)=(c1+c2,r1+r2) and (c1,r1)·(c2,r2)=(c1c2,c1r2+r1c2+r1r2), then C(∝)R is a ring. The identity of C(∝)R is (1R,0). Such ring is called the subring-extension of R. In particular, when the subring C is R, R(∝)R is called trivial subring-extension of R. The paper provided some relevant properties and examples to investigate the morphic properties of the subring-extension of R. It is shown that if S=C(∝)R is a left morphic ring, so are C and R and if R is a semisimple ring, then R(∝)R is a strongly morphic ring.