Section 7.2
Constructions with rings
Let (R,+,*,0,1) and (R',+',*',0',1') be rings.
Just like the product of two monoids (respectively, groups) is a
monoid (respectively, group), the product of two rings is a ring.
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Theorem
The Cartesian product R × R' with coordinatewise addition and multiplication and with zero element (0,0') and unit element (1,1') is a ring.
The ring is called the direct product of R and R' and often denoted by R × R' (instead of the full information with multiplication, addition, zero, and unit).
The Chinese remainder theorem can be nicely phrased in terms of direct products: If m and n are positive integers greater than 1 with gcd(m,n) = 1, then Z/mnZ is isomorphic with Z/mZ × Z/nZ; the isomorphism is given by the map
Hence, given an element x = (b mod m, c mod n) in Z/mZ × Z/nZ there is a unique element in Z/mnZ that is mapped onto x.
Theorem
(R × R')* =
R* × R'*.