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

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).

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
Canonical name	IsomorphismTheoremsOnAlgebraicSystems
Date of creation	2013-03-22 16:45:28
Last modified on	2013-03-22 16:45:28
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	8
Author	CWoo (3771)
Entry type	Theorem
Classification	msc 08A05