Ideals of Rings and Fields

An 'Ideal' is a very special class of subring. Ideals are especially useful in studying the structure of rings, particularly in homomorphisms (mappings) from one ring to another. If you choose to read any text on groups, rings and fields, an ideal is the ring analogue of a 'Normal Subgroup' in group theory.

Other than an ideal satisfying all of the ring axioms as a subset of a ring, there is one other condition.

A subring of a ring R is an Ideal 'I', if for all elements r in R and i in I, the products ri and ir are also elements of I.

This is more colloquially written as rI and Ir are in I..

Clearly, if unity belongs to the ideal I, then every element of R is in I, so I = R.

Ijn the case of Fields, since unity is in every subfield of a field, the only subfields of a field F that are ideals are F itself and the null Ring, {0}. These are often called the "trivial ideals".

Since rI = Ir = I, Ideals are useful as the Kernels of homomorphisms, which is the set of a ring R that is mapped onto the zero element of the Ring image 'S' of the Homomorphism, When the kernel is an ideal of R  the image S is a ring itself. The condition for I to be an ideal is comparable to the rule that 0x = x0 = 0, as it does in S. We consider the operations of S to be inherited from those of R, but "modulo the ideal I", or simply, "mod I".

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