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14. IDEALS AND FACTOR RINGS
Factor Rings
Theorem (14.2 — Existence of Factor Rings). Let R be a ring and A a
subring of R. The set of cosets {r+A|r 2 R} is a ring under the operations
(s + A) + (t + A) = s + t + A and (s + A)(t + A) = st + A ()
A is an ideal of R.
[In this case , we say R/A is a factor ring of R.]
Proof.
We know the set of cosets form a group under addition. If our multiplication
is well-defined, i.e., multiplication is a binary operation, it is clear that the
multiplication is associative and distributive over addition.
[To show multiplication is well-defined () A is an ideal of R.]
((=) Suppose A is an ideal of R and let s + A = s0 + A and t + A = t0 + A.
Now s = s0 + a and t = t” + b where a, b 2 A. Then
st = (s0 + a)(t0 + b) = s0t0 + s0b + at0 + ab =)
st + A = s0t0 + s0b + at0 + ab + A = s0t0 + A
since s0b + at0 + ab 2 A. Thus multiplication is well-defined.
(=)) (using contrapositive) Suppose A is a subring of R that is not an ideal.
then 9 a 2 A and r 2 R 3 ar 62 A or ra 62 A. WLOG, assume ar 2
6 A.
Consider a + A = 0 + A and r + A.
(a + A)(r + A) = ar + A, but (0 + A)(r + A) = 0 · r + A = A 6= ar + A,
so multiplication is not well-defined and R/A is not a ring.
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