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Thus any commutative local ring with Krull dimension zero is Jacobson, but if the Krull dimension is 1 or more, the ring cannot be Jacobson. (Amitsur 1956) showed that any countably generated algebra over an uncountable field is a Jacobson ring. Tate algebras over non-archimedean fields are Jacobson rings.
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 ...
If R is commutative, the Jacobson radical always contains the nilradical. If the ring R is a finitely generated Z-algebra, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equal to the intersection of all the maximal ideals of R that contain I. This says that R is a Jacobson ring.
In fact, if M is finitely generated over a ring, then rad(M) itself is a superfluous submodule. This is because any proper submodule of M is contained in a maximal submodule of M when M is finitely generated. A ring for which rad(M) = {0} for every right R-module M is called a right V-ring. For any module M, rad(M/rad(M)) is zero.
Every ring that is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself. A J-semisimple ring is semisimple if and only if it is an artinian ring, so semisimple rings are often called artinian semisimple rings to avoid confusion. For example, the ring of ...
In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. Primitive ideals are prime. The quotient of a ring by a left primitive ideal is a left primitive ring.
A ring R is said to act densely on a simple right R-module U if it satisfies the conclusion of the Jacobson density theorem. [7] There is a topological reason for describing R as "dense". Firstly, R can be identified with a subring of End( D U ) by identifying each element of R with the D linear transformation it induces by right multiplication.
The ring of integers is semiprimitive, but not semisimple. Every primitive ring is semiprimitive. The product of two fields is semiprimitive but not primitive. Every von Neumann regular ring is semiprimitive. Jacobson himself has defined a ring to be "semisimple" if and only if it is a subdirect product of simple rings, (Jacobson 1989, p. 203