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Problem set 6. - Mathematics TU Graz
Problem set 6. - Mathematics TU Graz

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Lecture 1. Modules

PRIME RINGS SATISFYING A POLYNOMIAL IDENTITY is still direct
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... own lower nil radical. We remark that a prime ideal, in fact any ideal modulo which there are no nilpotent ideals, is an algebra ideal, so that a prime quotient R of A is also a polynomial identity algebra in which every element is a sum of nilpotents. But by the last part of the theorem, (Pi = Ç), ...
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... elements in Q, we just need to figure out when the multiplicative inverse is contained in R. If a/b ∈ Q is nonzero, then (a/b)−1 = b/a. Therefore R× = {a/b ∈ R : b/a ∈ R}. Writing a/b in reduced form, we must have that both a and b are odd. Therefore R× is the multiplicative group of fractions whose ...
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1 Principal Ideal Domains

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Garrett 11-04-2011 1 Recap: A better version of localization...

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... Let I, J be ideals of a ring R such that I + J = R. Prove that I  J = IJ and if IJ = 0, then R ; R  R . ...
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Math 113 Final Exam Solutions

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... and we may form Rp = S −1 R = { as : a ∈ A, s ∈ S} addition and multiplication making sense because we can find common denominators. This process is called localization at p because pRp is the unique maximal ideal in Rp . Example 1.2. (a) Any field k is a local ring, since the only proper ideal (0) ...
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2.1 Modules and Module Homomorphisms

... A-modules correspond to ring homomorphisms from A into endomorphism rings of abelian groups. Examples: (1) If I  A then I becomes an A-module by regarding the ring multiplication, of elements of A with elements of I , as scalar multiplication. ...
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Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra.Some specific kinds of commutative rings are given with the following chain of class inclusions: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields
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