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There is an analogous list for one-sided ideals, for which only the right-hand versions will be given. For a right ideal A of a ring R, the following conditions are equivalent to A being a maximal right ideal of R: There exists no other proper right ideal B of R so that A ⊊ B. For any right ideal B with A ⊆ B, either B = A or B = R.
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A right ideal is defined similarly, with the condition replaced by . A two-sided ideal is a left ideal that is also a right ideal. If the ring is commutative, the three definitions are the same, and one talks simply of an ideal. In the non-commutative case, "ideal" is often used instead of "two-sided ideal".
A theorem of Hyman Bass in now known as "Bass' Theorem P" showed that the descending chain condition on principal left ideals of a ring R is equivalent to R being a right perfect ring. D. D. Jonah showed in ( Jonah 1970 ) that there is a side-switching connection between the ACCP and perfect rings.
In algebra and algebraic geometry, given a commutative Noetherian ring and an ideal in it, the n-th symbolic power of is the ideal = (/) ()where is the localization of at , we set : is the canonical map from a ring to its localization, and the intersection runs through all of the associated primes of /.
Americanism, also referred to as American patriotism, is a set of patriotic values which aim to create a collective American identity for the United States that can be defined as "an articulation of the nation's rightful place in the world, a set of traditions, a political language, and a cultural style imbued with political meaning". [1]
In rings with unity, minimal right ideals are necessarily principal right ideals, because for any nonzero x in a minimal right ideal N, the set xR is a nonzero right ideal of R inside N, and so xR = N. Brauer's lemma: Any minimal right ideal N in a ring R satisfies N 2 = {0} or N = eR for some idempotent element e of R (Lam 2001, p. 162). If N ...