Let R,S be two (commutative) rings. A function f: R -> S is this if it respects the ring operations. So f(a+b) = f(a) + f(b) and f(ab) = f(a)f(b)
Ring Homomorphism
1 of 9
Let U≠Ø be a subset of a ring (R,+,*). Then (U,+,*) is subring of (R,+,*) if and only if the following condititions hold: For all x,y∈U, x+(-y)∈U, and for all x,y∈U, x*y∈U
Subring Test
2 of 9
A subring I of a ring R is called an _____ if it satisfies RI c I, that is, ri∈I for all r∈R and all i∈I
Ideal
3 of 9
Any subring of a ring which consists of all multiples of a single element a∈R is an _________ ideal.
Principal
4 of 9
We call an integral domain this if all ideals are principal.
Principal Ideal Domain
5 of 9
A homomorphism f: R -> S is this if it is a bijection. (surjective and Ker f = {0})
Isomorphism
6 of 9
Let IcE be an ideal in R. Define this by a-b∈I (only the fact that I is a subring is needed)
Equivalence Relation
7 of 9
Let R/I denote the set of this (this is denoted [a] where b∈[a] if and only if a-b∈I)
Equivalence classes
8 of 9
Let f: R -> S be a ring homomorphism with kernal K. Then R/K≅Im f
Isomorphism Theorem
9 of 9
Other cards in this set
Card 2
Front
Let U≠Ø be a subset of a ring (R,+,*). Then (U,+,*) is subring of (R,+,*) if and only if the following condititions hold: For all x,y∈U, x+(-y)∈U, and for all x,y∈U, x*y∈U
Back
Subring Test
Card 3
Front
A subring I of a ring R is called an _____ if it satisfies RI c I, that is, ri∈I for all r∈R and all i∈I
Back
Card 4
Front
Any subring of a ring which consists of all multiples of a single element a∈R is an _________ ideal.
Back
Card 5
Front
We call an integral domain this if all ideals are principal.
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