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Let f be a function defined on a set A and taking values in a set B. Then f is said to be a surjection (or surjective map) if, for any b in B, there exists an a in A for ...

A morphism f:Y->X in a category is an epimorphism if, for any two morphisms u,v:X->Z, uf=vf implies u=v. In the categories of sets, groups, modules, etc., an epimorphism is ...

A general reciprocity theorem for all orders which covered all other known reciprocity theorems when proved by E. Artin in 1927. If R is a number field and R^' a finite ...

A transformation which is one-to-one and a surjection (i.e., "onto").

Let f be a function defined on a set A and taking values in a set B. Then f is said to be an injection (or injective map, or embedding) if, whenever f(x)=f(y), it must be the ...

Let K be a field, and A a K-algebra. Elements y_1, ..., y_n are algebraically independent over K if the natural surjection K[Y_1,...,Y_n]->K[y_1,...,y_n] is an isomorphism. ...

The term endomorphism derives from the Greek adverb endon ("inside") and morphosis ("to form" or "to shape"). In algebra, an endomorphism of a group, module, ring, vector ...

There are two possible definitions: 1. Possessing similarity of form, 2. Continuous, one-to-one, in surjection, and having a continuous inverse. The most common meaning is ...

The invertible matrix theorem is a theorem in linear algebra which gives a series of equivalent conditions for an n×n square matrix A to have an inverse. In particular, A is ...

The tensor product between modules A and B is a more general notion than the vector space tensor product. In this case, we replace "scalars" by a ring R. The familiar ...