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10 Cards in this Set
- Front
- Back
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What are the axioms for a Ring |
Addition Internal Associative Identity Inverse Multiplication Internal Associative Identity Inverse Distributativity Right Left |
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Define a Ring with one |
A Ring R is a Ring with one if it contains a multiplicative identity |
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Define a commutative ring |
A ring R is a commutative ring if multiplication is commutative within it |
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Define a zero divisor, a, in a commutative ring R |
a is a zero divisor if a =/= 0 and there exists b =/= 0 such that ab=0 |
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Define a unit, a, in a ring R |
a is a unit if there exists a⁻¹∈R such that aa⁻¹=1=a⁻¹a |
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Define a field F from the definition of a commutative ring R |
R is a field if it also contains multiplicative inverses |
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Let R be a commutative ring with one and a∈R with a=/=0. Suppose a is a unit. Then |
a is not a zero divisor |
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Let n∈ℕ with n>1. Then what condition on n is Zn an integral domain |
n is prime |
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State the division theorem for polynomials |
Let f(X), g(X)∈F[X] with g(X) =/= 0. Then there exist unique q(X),r(X)∈F[X] such that f(X) = q(X)g(X) + r(X), and r(X) =0 or deg[r(X)] < deg[g(X)] |
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State the first subring test |
Let R be a ring and S be a subset of R. Then S is a subring of R provided (SR1) 0∈S (SR2) for all x,y∈S, we have that x-y,xy∈S |
