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isomorphism theorems on algebraic systems
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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 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
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 $f':A/\mathfrak{C} \to (A/\mathfrak{C})/(\mathfrak{D}/\mathfrak{C})$ satisfying the equation $f=[\cdot]_{\mathfrak{D}/\mathfrak{C}}\circ f'$ . In other words, $$(A/\mathfrak{C})/(\mathfrak{D}/\mathfrak{C})\cong A/\mathfrak{D}.$$
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Cross-references: equation, induced, quotient algebras, onto, projections, congruences, extension, restricted, congruence, homomorphism, algebra, theorems, isomorphism, type, algebraic systems
This is version 5 of isomorphism theorems on algebraic systems, born on 2007-02-25, modified 2007-03-03.
Object id is 8984, canonical name is IsomorphismTheoremsOnAlgebraicSystems.
Accessed 764 times total.
Classification:
| AMS MSC: | 08A05 (General algebraic systems :: Algebraic structures :: Structure theory) |
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Pending Errata and Addenda
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![$\displaystyle \xymatrix{ & A \ar[dl]_{[\cdot]_{\mathfrak{C}}} \ar[dr]^{[\cdot]_{\mathfrak{D}}} & \ A/\mathfrak{C} \ar[rr]^f && A/\mathfrak{D} } $](http://images.planetmath.org:8080/cache/objects/8984/js/img1.png "$\displaystyle \xymatrix{ & A \ar[dl]_{[\cdot]_{\mathfrak{C}}} \ar[dr]^{[\cdot]_{\mathfrak{D}}} & \ A/\mathfrak{C} \ar[rr]^f && A/\mathfrak{D} } $")