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Any finitely generated algebra over a Jacobson ring is a Jacobson ring. In particular, any finitely generated algebra over a field or the integers, such as the coordinate ring of any affine algebraic set, is a Jacobson ring. A local ring has exactly one maximal ideal, so it is a Jacobson ring exactly when that maximal ideal is the only prime ideal.
The following characterization of a Jacobson ring contains Zariski's lemma as a special case. Recall that a ring is a Jacobson ring if every prime ideal is an intersection of maximal ideals. (When A is a field, A is a Jacobson ring and the theorem below is precisely Zariski's lemma.)
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R. [ 1 ] The theorem can be applied to show that any primitive ring can be viewed as a "dense" subring of the ring of linear transformations of a vector space.
More generally, one can construct the free algebra R E on any set E of generators. Since rings may be regarded as Z-algebras, a free ring on E can be defined as the free algebra Z E . Over a field, the free algebra on n indeterminates can be constructed as the tensor algebra on an n-dimensional vector space.
Let F be a free ring (that is, free algebra over the integers) with the set X of symbols, that is, F consists of polynomials with integral coefficients in noncommuting variables that are elements of X. A free ring satisfies the universal property: any function from the set X to a ring R factors through F so that F → R is the unique ring ...
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]
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