Our first naïve attempt is to take a ring quotient *R*/*S* for a subring *S* of *R*. First, (*S*, +) is a subgroup of (*R*, +) so we can denote every coset *R*/*S* by *x*+*S* for some *x* in *R*. Naturally we’ll want our 1+*S* to be the identity element in *R*/*S*, which immediately gives us a problem: since 1 is in *S*, 1+*S* = 0+*S* and we have 1=0 in *R*/*S*. This means *r’* = *r’*×1 = *r’*×0 = 0 for any *r’* in *R*/*S* which is ridiculous.

This forces us to abandon our approach. Instead we look at *subrngs* of *R*. Let *I* be any subrng. Once again, every coset *R*/*I* is represented by *x*+*I* for some *x.* In order for multiplication to be well-defined in *R*/*I*, we need the following condition:

,

i.e.

But *xy – x’y’* = *x*(*y* – *y’*) + (*x* – *x’*)*y’* and a moment of reflection tells us that our condition above is equivalent to the following:

if , , then .

Hence we define an

idealto be a subsetIofRsuch that:

Icontains 0;- if
x,ylies inI, thenx+ylies inI;- if
xlies inIandylies inR, thenxyandyxlie inI.Sometimes, one also says I is a

two-sided ideal; the reason for this will become apparent when we think of ideals as a special case of modules.

*Note: from the conditions above, it follows that I is a subgroup of (R, +); indeed to show that I is closed under subtraction, write x-y = x + (-1)y and apply the last property.*

–

*Examples of ideals*

- In
**Z**, the only ideals are*n***Z**where*n*is a non-negative integer. - In a field
*K*, the only ideals are {0} and*K*. [ Indeed, if*I*is an ideal containing some non-zero element*x*, then*x*has an inverse*y*; since*xy*= 1, we see that*I*contains 1 and hence every element of*K*. ] - In a commutative ring
*R*, for any , we can take the set of all multiples of*x*. Clearly, this is non-empty, closed under addition, and closed under multiplication by any element of*R*. - Recall that the set of upper-triangular real
*n*×*n*matrices forms a ring. The set of strictly upper-triangular matrices (whose diagonal entries are 0) forms an ideal. - The ring
*M*(*n*,**R**) of*n*×*n*real matrices forms a ring with no non-trivial ideals. [*Sketch of proof : let x be a non-zero matrix in ideal I. Left- and right-multiply by appropriate matrices such that the resulting matrix has only 1 non-zero entry….*]

In example 5, the ring in question has no ideals other than 0 and itself. We call such a ring a **simple** ring (just like a person who has no ideals is a simple person).