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For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals. Jacobson rings were introduced independently by Wolfgang Krull (1951, 1952), who named them after Nathan Jacobson because of their relation to Jacobson radicals, and by ...
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.
By the Jacobson Density characterization, a left full linear ring R is always left primitive. When dim D V is finite R is a square matrix ring over D , but when dim D V is infinite, the set of finite rank linear transformations is a proper two-sided ideal of R , and hence R is not simple.
The Jacobson radical of any field, any von Neumann regular ring and any left or right primitive ring is {0}. The Jacobson radical of the integers is {0}. The Jacobson radical of the integers is {0}. If K is a field and R is the ring of all upper triangular n -by- n matrices with entries in K , then J( R ) consists of all upper triangular ...
The Jacobson density theorem determines the structure of primitive rings; ... The concept of the Jacobson radical of a ring; that is, the intersection of all right ...
Let R be a commutative ring. By definition, a primitive ideal of R is the annihilator of a (nonzero) simple R-module. The Jacobson radical = of R is the intersection of all primitive ideals. Equivalently,
Let A be a ring and the set of all primitive ideals of A. Then there is a topology on Prim ( A ) {\displaystyle \operatorname {Prim} (A)} , called the Jacobson topology , defined so that the closure of a subset T is the set of primitive ideals of A containing the intersection of elements of T .
A primitive idempotent of a ring R is a nonzero idempotent a such that aR is indecomposable as a right R-module; that is, such that aR is not a direct sum of two nonzero submodules. Equivalently, a is a primitive idempotent if it cannot be written as a = e + f , where e and f are nonzero orthogonal idempotents in R .
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