Roth定理及一类方程的求解在环上的推广

Roth's Theorems and Solving an Equation over Rings

设Ｒ是一个含幺环且它的中心包含一个域，本文将Ｒｏｔｈ的等价与相似定理推广到了Ｒ上，研究了含幺正则环上的一类方程ａｘｂ＋ｃｙｄ＝ｅ有解的充要条件及其通解表达式．

Let R be a ring with identity which center contains a field. In this paper we prove that Roth's equivalence theorem and similarity theorem are valid over R, dedve a necessary and sufficient condition for the existence of and the expression for the general solution of the equation axb+cyd=e over a regular ring with identity.