Perfect ringIn the area of abstract algebra known as ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exist rings which are perfect on one side but not the other. Perfect rings were introduced in Bass's book. A semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left-right symmetric.
Morita equivalenceIn abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like R, S are Morita equivalent (denoted by ) if their are equivalent (denoted by ). It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958. Rings are commonly studied in terms of their modules, as modules can be viewed as representations of rings.
Serial moduleIn abstract algebra, a uniserial module M is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M, either or . A module is called a serial module if it is a direct sum of uniserial modules. A ring R is called a right uniserial ring if it is uniserial as a right module over itself, and likewise called a right serial ring if it is a right serial module over itself.
Invariant basis numberIn mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over R have a well-defined rank. In the case of fields, the IBN property becomes the statement that finite-dimensional vector spaces have a unique dimension. A ring R has invariant basis number (IBN) if for all positive integers m and n, Rm isomorphic to Rn (as left R-modules) implies that m = n.
Primitive idealIn mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. 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. The primitive spectrum of a ring is a non-commutative analog of the prime spectrum of a commutative ring.
Nilpotent idealIn mathematics, more specifically ring theory, an ideal I of a ring R is said to be a nilpotent ideal if there exists a natural number k such that I k = 0. By I k, it is meant the additive subgroup generated by the set of all products of k elements in I. Therefore, I is nilpotent if and only if there is a natural number k such that the product of any k elements of I is 0. The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings.
Semiprime ringIn ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical ideals and semiprime rings are the same as reduced rings. For example, in the ring of integers, the semiprime ideals are the zero ideal, along with those ideals of the form where n is a square-free integer. So, is a semiprime ideal of the integers (because 30 = 2 × 3 × 5, with no repeated prime factors), but is not (because 12 = 22 × 3, with a repeated prime factor).
Idempotent (ring theory)In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that a = a2 = a3 = a4 = ... = an for any positive integer n. For example, an idempotent element of a matrix ring is precisely an idempotent matrix. For general rings, elements idempotent under multiplication are involved in decompositions of modules, and connected to homological properties of the ring.
Primitive ringIn the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero. A ring R is said to be a left primitive ring if it has a faithful simple left R-module. A right primitive ring is defined similarly with right R-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M.
Radical of an idealIn ring theory, a branch of mathematics, the radical of an ideal of a commutative ring is another ideal defined by the property that an element is in the radical if and only if some power of is in . Taking the radical of an ideal is called radicalization. A radical ideal (or semiprime ideal) is an ideal that is equal to its radical. The radical of a primary ideal is a prime ideal. This concept is generalized to non-commutative rings in the Semiprime ring article.
