Every ring is a subring of a ring with unity


Let A be a ring (possibly not commutative, and possibly without unity). Then the set Z×A forms a ring under the operations

(m,X) + (n,Y) = (m+n, X+Y)
(m,X) × (n,Y) = (mn, mY + nX + XY)

This ring has unity (1,0) (where 1 is the integer 1, and 0 is the additive identity of A) and it contains the subring {(0,X) : X ∈ A} which is isomorphic to A.

(Note that there are seven operations at work here: the usual addition and multiplication on Z; the addition and multiplication on A giving it its ring structure; the addition and multiplication on Z×A defined above; and the operation of multiplying an integer by an element of A, defined in terms of the addition on A as the (unique) operation satisfying

1X = X and (n+1)X = nX + X

for all X in A and all n in Z.)

2006 December 27


Mail Steven: steven@amotlpaa.org