Commutative Algebra/Fractions, annihilator, quotient ideals

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.  
  2.  
  3.  
  4.   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  .