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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.
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.
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 ...
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".
J(R) is the intersection of all the right (or left) primitive ideals of R. J(R) is the maximal right (or left) quasi-regular right (resp. left) ideal of R. As with the nilradical, we can extend this definition to arbitrary two-sided ideals I by defining J(I) to be the preimage of J(R/I) under the projection map R → R/I.
Any field is a Jacobson ring. Any principal ideal domain or Dedekind domain with Jacobson radical zero is a Jacobson ring. In principal ideal domains and Dedekind domains, the nonzero prime ideals are already maximal, so the only thing to check is if the zero ideal is an intersection of maximal ideals.
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.
The fact that the Jacobson radical can be viewed as the intersection of all maximal right (left) ideals in the ring, shows how the internal structure of the ring is reflected by its modules. 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 ...