Algebra II - Rings Flashcards
Define a ring
What are zero divisors
If a and b are non-zero elements of a ring R with ab=0 then a and b are called zero divisors
Define a unit
An element a of a ring R is called a unit if it has a two-sided inverse under multiplication. there exists b in R such that ab = ba = 1
Define a division ring
A non-zero ring R is called a division ring if R \ {0} is a group under multiplication - all non zero elements are units
Define a subring
A subset S of a ring R is called a subring of R if it forms a ring under the same operations as R with the same identity element
Define an isomorphism between two rings
State the chinese remainder theorem
The rings Zm x Zn and Zmn are isomorphic if and only if m and n are coprime
Prove the chinese remainder theorem
Find an integer x with x = 5 mod 8 and x = 6 mod 19
Define a ring homomorphism
Define an ideal
How does addition for cosets work
(I + a1) + (I + a2) = I + (a1 + a2)
State the first isomorphism theorem for rings
When is an ideal maximal
Define a Euclidean domain
Theorem: Any Euclidean domain is a principal ideal domain
What two statements are equivalent to x|y for all x,y in R
y is in (x)
(y) is a subset of (x)
For x,y in R, when are x and y associates
x divides y and y divides x
What two statements are equivalent to x is an associate y
(y) = (x)
there exists a unit q in R with x = qr
Let r be in R \ {0}, when is r irreducible and when is r prime
Proposition: If R is a domain, then any prime element of R is irreducible
Proposition: If R is a principal ideal domain, then any irreducible element of R is prime
When is a complex number algebraic and transcendental
Define a UFD
Proposition: If R is a UFD, then every irreducible element of R is prime
In an integral domain R what is the link between irreducibles in R and R[x]?
they’re the same
Define a primitive polynomial
State Gauss’ lemma
A primitive irreducible polynomial in Z[x] remains irreducible in Q[x]
State Eisenstein’s criterion and prove it
