Rings, ideals, integral domains, polynomial rings, field extensions, finite fields and an intro to Galois theory.
41 cards · basic cards · AI-written, checked twice. Edit anything.
- What is a ring (R, +, ·)?
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An abelian group (R, +) together with an associative binary operation · such that · distributes over +- What is a commutative ring?
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A ring where multiplication is commutative: a·b = b·a for all elements- What is a ring with unity?
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A ring that contains a multiplicative identity element 1 such that 1·a = a·1 = a for all a- Define a zero divisor in a ring R.
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A nonzero element a such that there exists nonzero b with a·b = 0 or b·a = 0- What is a unit in a ring R?
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An element a with a multiplicative inverse, that is, an element b with a·b = b·a = 1- Define an integral domain.
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A commutative ring with unity and no zero divisors- What is an ideal I in a ring R?
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A nonempty subset I closed under subtraction and absorbing ring multiplication: a in I and r in R implies r·a and a·r are in I- What is a principal ideal?
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An ideal generated by a single element a, denoted (a) or aR, consisting of all elements of the form a·r- Define a prime ideal P in a commutative ring.
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An ideal P such that whenever a·b is in P, at least one of a or b is in P- What is the ideal generated by a set S?
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The smallest ideal containing S, consisting of all finite sums of elements of the form r·s·t where r, t are in R and s is in S- What is the quotient ring R/I?
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The ring of cosets a + I with operations (a+I)+(b+I) = (a+b)+I and (a+I)(b+I) = ab+I- What is a ring homomorphism f: R to S?
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A function preserving ring structure: f(a+b) = f(a)+f(b), f(a·b) = f(a)·f(b), and if R and S have unity, f(1_R) = 1_S- What is the kernel of a ring homomorphism f: R to S?
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The set ker(f) = {a in R : f(a) = 0_S}, which forms an ideal of R- State the First Isomorphism Theorem for Rings.
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If f: R to S is a ring homomorphism, then R/ker(f) is isomorphic to im(f)- What is the Chinese Remainder Theorem for Rings?
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If I_1, ..., I_n are pairwise coprime ideals (I_i + I_j = R for i ≠ j), then R/(I_1 ∩ ... ∩ I_n) ≅ R/I_1 × ... × R/I_n
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