Does there exist a ring R with a nonzero maximal ideal M such that R^2=R and MR = RM = 0?

Here R is associative but does not have an identity (obviously). It seems a simple enough question but I'm not even sure of the answer if I insist that M be the only proper ideal. I know R can't be commutative and that the conditions can't be relaxed (much); for example, there is a commutative ring satisfying these conditions if M need not be maximal.