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2. Jacobson ring - Wikipedia

   en.wikipedia.org/wiki/Jacobson_ring

   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.

3. Jacobson density theorem - Wikipedia

   en.wikipedia.org/wiki/Jacobson_density_theorem

   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.

4. Jacobson's conjecture - Wikipedia

   en.wikipedia.org/wiki/Jacobson's_conjecture

   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]

5. Zariski's lemma - Wikipedia

   en.wikipedia.org/wiki/Zariski's_lemma

   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.)

6. Glossary of commutative algebra - Wikipedia

   en.wikipedia.org/.../Glossary_of_commutative_algebra

   J-1 ring A J-1 ring is a ring such that the set of regular points of the spectrum is an open subset. J-2 ring A J-2 ring is a ring such that any finitely generated algebra is a J-1 ring. Jacobian 1. The Jacobian matrix is a matrix whose entries are the partial derivatives of some polynomials. 2.

7. Radical of a module - Wikipedia

   en.wikipedia.org/wiki/Radical_of_a_module

   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.