Morphisms / Factor Rings

Morphism is a term that covers many variations. The three most common are;


There are three 'Isomorphism theorems' for rings and their ideals. Before giving them, I'd like to clarify what is meant by doing arithmetic modulo an ideal.

An ideal I of a ring R is an additive subgroup of (R , +) and it's order divides that of R by Lagrange's Theorem. We can split R into equivalence classes additively in the form  r + I  (since I is closed, elements of I form the identity  0 + I ). Addition is as simple as one would expect, multiplication is well defined also as,

(r + I)(s + I) = rs + rI + sI + rsII = rs + (r + s + rsI)I = rs + I

The sub-structure of R defined mod I, is named "The Factor Ring"  R/I.

A Homomorphism φ on a ring R satisfies for all elements r and s of R,
φ( r + s ) = φ( r ) + φ( s )

φ( rs ) = φ( r ) * φ( s )
where * is the multiplication of the image of φ( R )

The first Isomorphism theorem states that any homomorphism φ on a ring R to a Ring S gives rise to a natural isomorphism between;

 R / ker(φ) and S,  (that an ideal I is ker(φ)) 

where ker(φ) is the kernel of the map φ. An isomorphism is a one-to-one homomorphism. If two structures are algebraically identical even if they are described differently, they would be isomorphic is such a map exists.

The second Isomorphism relies on the fact that for two ideals I and J of a ring R:  

I + J = {i + j : i in I, j in J}
IJ = {ij : i in I, j in J} = I ∩ J (the intersection of I and J)

Are indeed both ideals.

The second isomorphism sates that there is an isomorphism between;

(I + J) / J and I / IJ

This can be generalised in the case that it will hold also if  I  is only a subring, with J an ideal, though the converse that;

(I + J) / I  "is isomorphic to"  J/IJ

will not be true.

The third isomorphism theorem for Rings states that for two ideals I and J of a ring R, with J a subring and ideal of I, there is an isomorphism between;

[R / J] / [I / J] and R / I

Because the subfield ideals of a field are trivial, the homomorphisms of a field onto it's subfields require the subfield to be trivial. Since the isomorphism of a field onto a possible rearrangement of itself is the only (interesting) type of homomorphism, these morphisms are termed "Automorphisms" .

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