Isomorphism Theorem

In the third section, we shall prove a Homomorphism Theorem and two Isomorphism Theorems for schemes of finite valency. All three of these results naturally generalize the finite versions of Emmy Noether s corresponding theorems for groups. 

The following theorem is called the First Isomorphism Theorem for schemes. 

Note that the Homomorphism Theorem and the First Isomorphism Theorem deal with factorizations over arbitrary closed subsets, whereas the Second Isomorphism Theorem deals with factorizations over normal closed subsets. 

We shall generalize the Second Isomorphism Theorem in the next section. 

The Homomorphism Theorem and the two Isomorphism Theorems were first proved in [35]. The thin case was already proved in 1929 by Emmy Noether cf. [32 I. 2],... 

The first section of this chapter provides general observations on modules over associative rings with 1. The collection includes the Homomorphism Theorem and the Isomorphism Theorem for modules over associative rings with 1. 

The following theorem is the Isomorphism Theorem for modules over associative rings with 1. 

Nicholson V, Tsai C-C, Johnson M, Naim M. A subgraph isomorphism theorem for molecular graphs. Stud Phys Theor Chem 1987 51 226-230. 

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