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In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one-sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily ...
A two-sided ideal is a left ideal that is also a right ideal. If the ring is commutative, the three definitions are the same, and one talks simply of an ideal. In the non-commutative case, "ideal" is often used instead of "two-sided ideal". If I is a left, right or two-sided ideal, the relation if and only if
If R is a non-commutative ring, J(R) is not necessarily equal to the intersection of all maximal two-sided ideals of R. For instance, if V is a countable direct sum of copies of a field k and R = End(V) (the ring of endomorphisms of V as a k-module), then J(R) = 0 because R is known to be von Neumann regular, but there is exactly one maximal ...
The Jacobson radical m of a local ring R (which is equal to the unique maximal left ideal and also to the unique maximal right ideal) consists precisely of the non-units of the ring; furthermore, it is the unique maximal two-sided ideal of R. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to ...
J(R) is the intersection of all the right (or left) primitive ideals of R. J(R) is the maximal right (or left) quasi-regular right (resp. left) ideal of R. As with the nilradical, we can extend this definition to arbitrary two-sided ideals I by defining J(I) to be the preimage of J(R/I) under the projection map R → R/I.
The above definition is satisfied if R has a finite number of maximal right ideals (and finite number of maximal left ideals). When R is a commutative ring, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many maximal ideals".
A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. Primitive ideals are prime. The quotient of a ring by a left primitive ideal is a left primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields.
Every two-sided ideal in a ring R is the kernel of some ring homomorphism. ... then f −1 (M) is a maximal ideal of R. If R and S are commutative and S is an ...