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In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good" elements of the ring. The first example of a radical was the nilradical introduced by Köthe (1930), based on a suggestion of Wedderburn (1908). In the next few years several other radicals were discovered, of which the most important example is the Jacobson ...
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 ...
Consider the ring of integers.. The radical of the ideal of integer multiples of is (the evens).; The radical of is .; The radical of is .; In general, the radical of is , where is the product of all distinct prime factors of , the largest square-free factor of (see Radical of an integer).
The socle of a ring R can refer to one of two sets in the ring. Considering R as a right R-module, soc(R R) is defined, and considering R as a left R-module, soc(R R) is defined. Both of these socles are ring ideals, and it is known they are not necessarily equal. If M is an Artinian module, soc(M) is itself an essential submodule of M.
The classical ring of quotients for any commutative Noetherian ring is a semilocal ring. The endomorphism ring of an Artinian module is a semilocal ring. Semi-local rings occur for example in algebraic geometry when a (commutative) ring R is localized with respect to the multiplicatively closed subset S = ∩ (R \ p i ) , where the p i are ...
The following equivalent definitions of a left perfect ring R are found in Anderson and Fuller: [2]. Every left R-module has a projective cover.; R/J(R) is semisimple and J(R) is left T-nilpotent (that is, for every infinite sequence of elements of J(R) there is an n such that the product of first n terms are zero), where J(R) is the Jacobson radical of R.
A ring R is called a Jacobson ring if the nilradical and Jacobson radical of R/P coincide for all prime ideals P of R. An Artinian ring is Jacobson, and its nilradical is the maximal nilpotent ideal of the ring. In general, if the nilradical is finitely generated (e.g., the ring is Noetherian), then it is nilpotent.
For example, in the ring of integers, the semiprime ideals are the zero ideal, along with those ideals of the form where n is a square-free integer. So, 30 Z {\displaystyle 30\mathbb {Z} } is a semiprime ideal of the integers (because 30 = 2 × 3 × 5, with no repeated prime factors), but 12 Z {\displaystyle 12\mathbb {Z} \,} is not (because 12 ...