Commutative Algebra/Fractions, annihilator, quotient ideals

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The quotient of two ideals[edit]

Definition 19.1:

Let be a ring and let be ideals. Then the quotient ideal (also written ) is defined to be

.

We note some properties:

Theorem 19.2 (properties of the quotient ideal):

Let be a ring and ideals.

  1. and, more generally,

The points 1. and 2. make calling those ideals "quotient" plausible, 3. and 4. less so (although the ideal still gets smaller when adding something to the denominator or shrinking the numerator).

Proof:

1.

2.

3.

where the middle equivalence follows since is the smallest ideal containing and , and thus is contained in every ideal where the latter two are contained.

4.

Definition 19.3:

In the case for an , we write

for .

Exercises[edit]

  • Exercise 19.1.1: Prove that for a ring and any ideal , and .