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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.
For noncommutative rings, the analogues for maximal left ideals and maximal right ideals also hold. For pseudo-rings, the theorem holds for regular ideals. An apparently slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows: Let R be a ring, and let I be a proper ideal of R.
For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements r ∈ R such that rM = 0 whenever M is a simple R-module.That is, = {=}. This is equivalent to the definition in the commutative case for a commutative ring R because the simple modules over a commutative ring are of the form R / for some maximal ideal of R, and the annihilators of R / in R are ...
The intersection of all maximal right ideals which are modular is the Jacobson radical. [8] Examples. In the non-unital ring of even integers, (6) is regular (=) while (4) is not. Let M be a simple right A-module. If x is a nonzero element in M, then the annihilator of x is a regular maximal right ideal in A.
If U is a right module over a ring, R, and I is a right ideal in R, then define U·I to be the set of all (finite) sums of elements of the form u·i, where · is simply the action of R on U. Necessarily, U·I is a submodule of U. If V is a maximal submodule of U, then U/V is simple.
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 terms of per-capita poverty, America is doing a little bit better than Costa Rica, but not quite as well as Romania. More: The Complicated Topic of Wage Inflation and the Controversy Surrounding It
It is also a fact that the intersection of all maximal right ideals in a ring is the same as the intersection of all maximal left ideals in the ring, in the context of all rings; irrespective of whether the ring is commutative. Noncommutative rings are an active area of research due to their ubiquity in mathematics.