- #1
hsetennis
- 117
- 2
Homework Statement
Let (R,+, ·) be an algebraic object that satisfies all the axioms for a ring
except for the multiplicative identity. Define addition and multiplication in
R × Z by
(a, n) + (b,m) = (a + b, n + m) and
(a, n) · (b,m) = (ab + ma + nb, nm).
Show that (R × Z,+, ·) is a ring that contains a subset in one-to-one
correspondence with R that has all the properties of the algebraic object
(R,+, ·).
Homework Equations
7 conditions for a ring: commute+, associate+, 1, 0, inverse+, associate×, distribute
The Attempt at a Solution
Trying to prove associativity, and I think I'm making some silly computational error, but I've been at it for hours and I can't catch what I'm missing:
[itex][(a,n)*(b,m)]*(c,p) = (ab+ma+nb,nm)*(c,) = (abc+mac+nbc+pab+pma+pnb+nmc,cp)[/itex]
while
[itex](a,n)*[(b,m)*(c,p)] = (a,n)*(bc+pb+mc,cp) = (abc+apb+amc+cpa+nbc+npb+nmc,bccp+pbcp+mccp)[/itex]