What are Quotient Spaces and How Are They Used in Algebra and Topology?

[flyerpower]

I'm having some troubles understanding the concepts of quotient algebra.
May someone explain me what exactly they are, giving some concrete examples?

I know that a quotient set is the set of all equivalence classes, but it sounds very vague for me and i can't make the analogy with quotient spaces, or quotient groups.

 

[Deveno]

what do you mean by "quotient algebra"? there are different kinds of algebras:

1. universal algebras (defined by a set and a signature of arity, and identities involving n-ary operations, etc.)
2. associative algebras (vector spaces with an associative ring structure).

 

[Fredrik]

what do you mean by "quotient algebra"?

If A is an algebra and I is an ideal in A, the quotient algebra A/I is defined in the following way. Define a relation ~ on A by saying that x~y if x-y is a member of I. It's easy to show that ~ is an equivalence relation. The equivalence class that contains x is denoted by x+I. The set of all equivalence classes is given the structure of an algebra by the definitions

(x+I)+(y+I)=(x+y)+I
a(x+I)=(ax)+I
(x+I)(y+I)=xy+I

 

[Deveno]

If A is an algebra and I is an ideal in A, the quotient algebra A/I is defined in the following way. Define a relation ~ on A by saying that x~y if x-y is a member of I. It's easy to show that ~ is an equivalence relation. The equivalence class that contains x is denoted by x+I. The set of all equivalence classes is given the structure of an algebra by the definitions

(x+I)+(y+I)=(x+y)+I
a(x+I)=(ax)+I
(x+I)(y+I)=xy+I

yes, that follows by definition 2. in the algebras of type 1, you generally don't have ideals, but rather congruences. i was asking "which" definition of algebra he meant.