Proper Ideal

Any ideal of a ring which is strictly smaller than the whole ring. For example, is a proper ideal of the ring of integers , since .

The ideal of the polynomial ring is also proper, since it consists of all multiples of , and the constant polynomial 1 is certainly not among them.

In general, an ideal of a unit ring is proper iff . The latter condition is obviously sufficient, but it is also necessary, because would imply that for all ,

so that , a contradiction.

Note that the above condition follows by definition: an ideal is always closed under multiplication by any element of the ring. The same property implies that an ideal containing an invertible element cannot be proper, because , where denotes the multiplicative inverse of in .

Since in field all nonzero elements are invertible, it follows that the only proper ideal of is the zero ideal.