On the primitivity of prime rings

Unions of chains of primes

“…To show that the ideal P n is primitive, it is enough by [7] to prove that e(R/P n )e is a primitive ring. We construct an isomorphism between e(R/P n )e and the countably generated free algebra F z 0 , z 1 , .…”

The theory of prime ideals of Leavitt path algebras over arbitrary graphs

“…By using Theorems 3.10 and 4.2, we are able to characterize the primitive ideals of L K (E) in the next theorem. We also need the well-known fact (see [19], Theorem 1) that a not-necessarily unital ring R is right (left) primitive if and only if there is an idempotent a ∈ R such that aRa is a right (left) primitive ring. Proof.…”

“…The Leavitt path algebras were introduced in [1] and [9] as algebraic analogs of the C * -algebras ( [19]) and the study of their algebraic structure has been the subject of a series of papers in recent years (see, for e.g., [1] - [13], [15], [22]). In this paper we develop the theory of the prime ideals of the Leavitt path algebras L K (E) for an arbitrary sized graph E. Here the graph E is arbitrary in the sense that no restriction is placed either on the number of vertices in E or on the number of edges emitted by a single vertex in E (such as row-finite or countable).…”

On Primitive Ideals in Polynomial Rings over Nil Rings

“…It is equivalent [4] It is known that if a polynomial ring R[x] is primitive then R need not be primitive [3] ( see also Bergman's example in [5]). Let R be a prime ring and I a nonzero ideal of R. Then R is a primitive ring if and only if I is a primitive ring [6]. Since the Hodges example has a nonzero Jacobson radical it follows that polynomial rings over Jacobson radical rings can be right and left primitive (see also Theorem 3).…”