Show briefly that 6Z is an ideal of 2Z Find addition and mul

Show briefly that 6Z is an ideal of 2Z. Find addition and multiplication table for the factor ring 2Z/6Z Is the factor ring in ) isomorphic to the field Z? Why?

Solution

6 2 2Z and 6 2 3Z, but 6 =2 pZ for any other prime p. In general, in a commutative ring
R, (b) (a) if and only if b 2 (a) if and only if a divides b (b = ra). So the containments between
principal ideals really re
ect the multiplicative structure. This is what algebraic number theory is
all about. (This is where the term \\ideal\" comes from - they were originally called \\ideal numbers\".
We can also look at any subset of a ring A R and look at the ideal generated by all the
elements in A.
If A is any subset of R, then
(A) = fr1a1s1 + : : : rnansn jri; si 2 R; ai 2 Ag
is an ideal, called the ideal generated by A.