# 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 $f:A\to B$ is a homomorphism from algebras $A$ and $B$. Then

 $A/\ker(f)\cong f(A).$
###### Theorem 2.

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

 $B/\mathfrak{C}_{B}\cong B^{\mathfrak{C}}/\mathfrak{C},$

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

###### Theorem 3.

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

1. 1.

there is a unique homomorphism $f:A/\mathfrak{C}\to A/\mathfrak{D}$ such that

 $\xymatrix{&A\ar[dl]_{[\cdot]_{\mathfrak{C}}}\ar[dr]^{[\cdot]_{\mathfrak{D}}}&% \\ A/\mathfrak{C}\ar[rr]^{f}&&A/\mathfrak{D}}$

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

2. 2.

Furthermore, if $ker(f)=\mathfrak{D}/\mathfrak{C}$, then

• $\mathfrak{D}/\mathfrak{C}$ is a congruence on $A/\mathfrak{C}$, and

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

 $(A/\mathfrak{C})/(\mathfrak{D}/\mathfrak{C})\cong A/\mathfrak{D}.$
Title isomorphism theorems on algebraic systems IsomorphismTheoremsOnAlgebraicSystems 2013-03-22 16:45:28 2013-03-22 16:45:28 CWoo (3771) CWoo (3771) 8 CWoo (3771) Theorem msc 08A05