Describe all ring homomorphisms from z to zxz and from zxz to z
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be a ring homomorphism of  into  . Since  ,  is determined by the values of  and  . Suppose that  . By the homomorphism property

, you have  , and hence  and  . It follows that and  . So, the value of  is one of the elements  ,  ,  , or  . Likewise, the value of  is also one of the elements  ,  ,  , or  .

, you have  , and hence  and  . It follows that and  . So, the value of  is one of the elements  ,  ,  , or  . Likewise, the value of  is also one of the elements  ,  ,  , or  .
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