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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 ...
Jacobson's conjecture has been verified for particular types of Noetherian rings: Commutative Noetherian rings all satisfy Jacobson's conjecture. This is a consequence of the Krull intersection theorem. Fully bounded Noetherian rings [4] [5] Noetherian rings with Krull dimension 1 [6] Noetherian rings satisfying the second layer condition [7]
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 ...
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 ...
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