isomorphism theorems on algebraic systems

In this entry, all algebraic systems are of the same type; they are all $O$-algebras. We list the generalizations of three famous isomorphism theorems, familiar to those who have studied abstract algebra in college.

Theorem 1.

If $f\mathrm{:}A\mathrm{\to }B$ is a homomorphism from algebras $A$ and $B$. Then

$$A/\ker (f)\cong f(A).$$

Theorem 2.

If $B\mathrm{\subseteq }A$ are algebras and $\mathrm{C}$ is a congruence (http://planetmath.org/CongruenceRelationOnAnAlgebraicSystem) on $A$, then

$$B/{ℭ}_B\cong B^{ℭ}/ℭ,$$

where ${\mathrm{C}}_B$ is the congruence restricted to $B$, and $B^{\mathrm{C}}$ is the extension of $B$ by $\mathrm{C}$.

Theorem 3.

If $A$ is an algebra and $\mathrm{C}\mathrm{\subseteq }\mathrm{D}$ are congruences on $A$. Then

1. there is a unique homomorphism $f:A/ℭ\to A/𝔇$ such that

   where the downward pointing arrows are the natural projections of $A$ onto the quotient algebras (induced by the respective congruences).

2. Furthermore, if $ker(f)=𝔇/ℭ$, then

   • $𝔇/ℭ$ is a congruence on $A/ℭ$, and

   • there is a unique isomorphism $f^{\prime }:A/ℭ\to (A/ℭ)/(𝔇/ℭ)$ satisfying the equation $f={[\cdot ]}_{𝔇/ℭ}\circ f^{\prime }$. In other words,

     $$(A/ℭ)/(𝔇/ℭ)\cong A/𝔇.$$