[Abstract Algebra] Chapter 4 Flashcards
What are the elements in a ring?
a set, an addition operation, a multiplication operation
The set and the addition operation in a ring form what type of structure?
an Abelian group
What property does the multiplication operation necessarily have?
associativity
What property does the multiplication operation most commonly have?
commutativity
What type of function is the multiplication operation?
binary function
What relation must the multiplication operation have with regards to the addition operation?
multiplication must distribute over addition
What is a zero ring?
A ring which consists of only one element
Rings (in the Napkin) must contain what two elements?
Additive identity and multiplicative identity
What is a product ring?
Ordered pairs (r, s), each element being from a different ring, where operations are done component-wise
What is a polynomial ring?
Sets of polynomials with all coefficients in ring R
How is the notation for a polynomial ring written?
R[x]
How is the notation for a polynomial ring pronounced?
R adjoin x
How many variables can a polynomial ring have?
Any number, including infinite
What is a unit?
Elements which have a multiplicative inverse
What is a field?
Non-trivial rings which all non-zero elements are units
How can a zero ring be identified using identity elements?
Additive and multiplicative identities are the same element
When is the ring Z/nZ a field?
When n is prime
What commonly used sets are fields?
Q, R, C
What criteria do ring homomorphisms must have in regards to addition?
f(r1) + f(r2) = f(r1 + r2)
What criteria do ring homomorphisms must have in regards to multiplication?
f(r1) * f(r2) = f(r1 * r2)
What criteria do ring homomorphisms must have in regards to identities?
f(1R) = 1S, where 1R is the multiplicative identity of R, etc.
What is a kernel of a ring homomorphism?
The set of elements which are mapped to 0S under the ring homomorphism
What two properties does the kernel of a ring homomorphism have?
Closed under addition, absorbs multiplication from any other element of R
What are sets satisfying the two properties of a kernel called?
Ideals
What is a proper ideal?
Ideals which are not equal to the ring it is part of
How can improper ideals be identified?
They contain a unit
How many ideals does a field contain?
Two; the zero ring and the entire field
How is the ideal generated by a set of elements, x_i, denoted?
(x_1, x_2, …, x_i, …, x_n)
What are ideals containing only one element called?
Principal ideals
What is a quotient ring?
{r + I| r in R} for an ideal I of R
What type of structure are the elements of a quotient ring?
Equivalence classes
How is a quotient ring denoted?
R/I
How is a quotient ring pronounced?
R mod I
What is a simple way to find elements of a quotient ring?
Assume any element of I is equal to zero, or “modding out” by elements of I
What is a principal ideal ring?
A ring where all ideals are principal ideals
For a field k, what ring related to k is always a principal ideal ring?
k[x]
Is the ideal (10, 15) principal?
Yes, as (10, 15) = 5
Why are brackets used to denote the greatest common factor between some amount of numbers?
As the ideal generated by those numbers is congruent to the principal ideal generated by their greatest common factor
Is the ideal (x, 2015) principal?
No, as any f(x) generating the ring must divide both x and 2015 which is impossible
What is a Noetherian ring?
Rings where every ideal can be generated with finitely many elements
What is an example of a non-Noetherian ring?
Z[x_1, x_2, x_3, …] has the ideal (x_1, x_2, x_3, …)
What condition regarding chains of ideals do only non-Noetherian rings not satisfy?
The existence of an infinite ascending chain of ideals I1 being a subset of I2, being a subset of I3, …
Given a Noetherian ring R, what related ring is always Noetherian?
R[x]
