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Proper Ideal


Any ideal of a ring which is strictly smaller than the whole ring. For example, 2Z is a proper ideal of the ring of integers Z, since 1 not in 2Z.

The ideal <X> of the polynomial ring R[X] is also proper, since it consists of all multiples of X, and the constant polynomial 1 is certainly not among them.

In general, an ideal I of a unit ring R is proper iff 1 not in I. The latter condition is obviously sufficient, but it is also necessary, because 1 in I would imply that for all a in R,

 a=a·1 in I,

so that I=R, 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 I containing an invertible element a cannot be proper, because 1=a^(-1)·a in I, where a^(-1) denotes the multiplicative inverse of a in R.

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


See also

Ideal, Maximal Ideal, Proper Subset, Ring

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Proper Ideal." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ProperIdeal.html

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