[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