show that R x Zn has an identity element.

Question

Suppose that R is a ring of characteristic n. If addition and multiplication are defined in R x Zn = {(x, a)lx E R; a E Zn} by
(x, a) + (y, b) = (x + y, a +n b),

(x, a)(y, b) = (xy + ay + bx, a·nb),
prove that R x Zn is an extension ring of R of characteristic n. Also show that R x Zn has an identity element.

Answer 1

You need only check the ring axioms for this new 'ring' R x Z_n. Notice these follow easily since the new ring is merely the original operations in R and Z_n in each component. This ring also clearly contains R via identifying it with the subring (R,0) (strictly speaking, it does not contain it but rather an isomorphic copy of it). Now given multiplication occurs component-wise, R has identity, and Z_n has identity, what do you think the identity is?