# 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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If I is a maximal ideal, then [??](I) = I which proves I is closed in C(X).
It is a local ring, and [M.sub.v] := {x [member of] K | v(x) > 0} is its maximal ideal. Let [[bar.K].sub.v] := [O.sub.v]/[M.sub.v] = {[bar.a] = a + [M.sub.v] | a [member of] [O.sub.v]} be the residue field (or [bar.K] when there is no danger of confusion).
Its unique maximal ideal consists of the zero element.
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.
Let [M.sub.x1] be a maximal ideal containing [[chi].sub.U].
under the double binary operation ([+.sub.1], [x.sub.1]), where [[??].sub.11] is a maximal ideal subspace of [??] and in general, for any integer i, 1 [less than or equal to] i [less than or equal to] m - 1, [[??].sub.1(i+1])] is a maximal ideal subspace of [[??].sub.1i].
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.
A test-ring A (or (A,[m.sub.A])) is a local k-algebra, whose maximal ideal [m.sub.A] is nilpotent with residue field A/[m.sub.A] [congruent to] k.
Let R be a discrete valuation ring with maximal ideal m = ([pi]) and finite residue field of order q := [absolute value of R/([pi]))].

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