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Glossary of ring theory - ... The relationship between ideals, homomorphisms, and factor rings is summed up in the fundamental theorem on homomorphisms. ; Finitely generated ideal : A left ideal I is finitely generated if there exist finitely many elements a1,...,an such that I = Ra1 + ... + Ran. A right ... sided ideal I is finitely generated if there exist finitely many elements a1,...,an such that I = Ra1R + ... + RanR. ; Ideal : A left ideal I of R is a subgroup or (R,+) such that aI I for all a R. A right ideal ... I for all a R. An ideal (sometimes for emphasis: a two-sided ideal) is a subgroup which is both a left ideal and a right ideal. ; Jacobson radical : The intersection of all maximal left ideals in a ring forms a ...

Local ring - ... functions defined on varieties or manifolds. Table of contents showTocToggle("show","hide") 1 Definition and first consequences 2 Examples 2.1 Commutative 2.2 Non-commutative 3 Some facts and defintions 3.3 Commutative 3.4 General Definition and first consequences A ring R is local if it has one (and therefore all) ...

Lie group - ... Lie group is connected. This is done as follows. A vector field on a Lie group G is said to be left-invariant if it commutes with left translation, which means the following. Define Lg[f](x)= f(gx) for any analytic function f : G -> F and ...

Adjoint functors - ... T, U with Tx,y> = x,Uy>, which is formally similar to the above Hom relation. We say that F is left adjoint to G, and G is right adjoint to F. Since G may have itself a right adjoint, quite different ... homomorphisms R -> S, with S a ring having a multiplicative identity. The morphisms in C must fill in triangles that are commutative diagrams, and preserve multiplicative identity. The assertion is that C has an initial object R -> R*, and R* is then ... assumption of multiplicative identity. There is a functor from C1 to C0 that forgets about the 1. We are seeking a left adjoint to it. This is a clear, if dry, formulation. One way to see what is achieved by using ...

Derived functor - ... long exact sequence". The concept of derived functors explains and clarifies many of these observations. Suppose we are given a covariant left exact functor F : A B between two abelian categories A and B. If is a short exact sequence in A ... there exists a monomorphism A I where I is an injective object in A. The right derived functors of the covariant left-exact functor F : A B is then defined as follows. Start with an object X of A. Because there are ... A of A there exists an epimorphism P A where P is a projective object), then one can define analogously the left-derived functors LiG. For an object X of A we first construct a projective resolution of the form where ...