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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.
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 ...
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.
2. A Jacobson ring is a ring such that every prime ideal is an intersection of maximal ideals. Japanese ring A Japanese ring (also called N-2 ring) is an integral domain R such that for every finite extension L of its quotient field K, the integral closure of R in L is a finitely generated R module.
The lemma may also be understood from the following perspective. In general, a ring R is a Jacobson ring if and only if every finitely generated R-algebra that is a field is finite over R. [2] Thus, the lemma follows from the fact that a field is a Jacobson ring.
Given Zariski's lemma, proving the Nullstellensatz amounts to showing that if k is a field, then every finitely generated k-algebra R (necessarily of the form = [,,] /) is Jacobson. More generally, one has the following theorem: Let be a Jacobson ring.
In algebra, ring theory is the study of rings, ... The concept of the Jacobson radical of a ring; that is, the intersection of all right (left) ...
2. An ideal I of a ring R is semiprime if for any ideal A of R, A n ⊆ I implies A ⊆ I. Equivalently, I is semiprime if and only if R/I is a semiprime ring. semiprimitive A semiprimitive ring or Jacobson semisimple ring is a ring whose Jacobson radical is zero. Von Neumann regular rings and primitive rings are semiprimitive, however quasi ...
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