# maximal ideal

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## maximal ideal

[¦mak·sə·məl ī′dēl]
(mathematics)
An ideal I in a ring R which is not equal to R, and such that there is no ideal containing I and not equal to I or R.
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References in periodicals archive ?
v] := {x [member of] K | v(x) > 0} is its maximal ideal.
Let P be a maximal ideal of Z[theta] and p the prime such that pZ = P [union] Z.
Our new rank functions depend on the ideal structure of the semirings and this leads us to study semirings which have a unique maximal ideal in Section 5.
For a less trivial example, let M be a unique maximal ideal of a near-ring N with [M.
i+1)1] is a maximal ideal subspace of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and in general, [[?
The finite hyperrationals constitute a valuation ring in the latter field, and the non-invertible members of the ring constitute (as in every valuation ring) a maximal ideal - in this case, they are the infinitesimal hyperrationals.
Now q is contained in some maximal ideal m of B, which lies over another maximal ideal n of A containing p.
x]) in the ring [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (t is here a generator of the maximal ideal [m.
Let R be a discrete valuation ring with maximal ideal m = ([pi]) and finite residue field of order q := [absolute value of R/([pi]))].
They proved two theorems as follow: In an ordered semigroup S, for which there exists an element a [member of] S such that the ideal of S generated by a is S, there is at most one maximal ideal which is the union of all proper ideals of S.
If m is the maximal ideal of R, a construction of Serre and Tate gives a pairing q: [T.
If L is not pseudocompact, then R x L has a free proper ideal, and hence a free maximal ideal since every free proper ideal is contained in a free maximal ideal.

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