110B Flashcards
What is a ring?
is a set R together with two binary operations +,• that follows these properties:
R1: is an abelian group
R2: • is associative
R3: for all a,b,c€R the left and right distributive properties hold
What are the 3 basic ring axioms?
1: 0a=a0=0
2: a(-b)=(-a)b=-(ab)
3: (-a)(-b)=ab
What is a homomorphism in terms of rings?
For rings R and R’ a map §:R->R’ is a homomorphism if the following conditions are satisfied for all a,b€R:
1: §(a+b)=§(a)+§(b)
2: §(ab)=§(a)§(b)
What is the evaluation homomorphism?
For F:the ring of all functions mapping R into R and for each a€R, the evaluation homomorphism is §a:F->R where §(f)=f(a) for f€F
What does it mean that two rings, R and R’, are isomorphic?
There is a homomorphism §:R->R’ that is 1-1 and onto
What is a commutative ring?
A ring in which multiplication is commutative
What is a ring with unity?
A ring that has a multiplicative identity. The identity element itself is called “unity”
What is a unit?
In a ring with unity, a unit is any element in the ring that has a multiplicative inverse.
What is a division ring?
A ring in which every nonzero element is a unit
What is a field?
A commutative division ring (commutative ring in which every nonzero element is a unit)
What is a zero divisor?
If a and b are nonzero elements of a ring such that ab=0 then a and b are zero divisors (or divisors of zero)
What are the zero divisors in Zn?
The nonzero elements that are not relatively prime to n
What are the zero divisors in Zp where p is a prime?
Zp has no zero divisors
When do the cancellation laws hold in a ring R?
When R has no zero divisors
What is an integral domain?
A commutative ring with unity that has no zero divisors
