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Tag Archives: ring theory
Topics in Commutative Rings: Unique Factorisation (3)
Example 1: The Gaussian Integers Z[i] Let’s pick the norm function N : Z[i]-{0} → N where N(a+bi) = (a+bi)(a–bi) = a2+b2. We know that N is a multiplicative function, i.e. N(r)N(s) = N(rs). Instead of checking this by brute force, we write N(x) = x·xc, where (a+bi)c = a-bi is the conjugate of a+bi. It’s easy to … Continue reading
Topics in Commutative Rings: Unique Factorisation (2)
In the previous article, we imposed certain finiteness conditions on the ring (specifically a.c.c. on principal ideals: that every increasing sequence of principal ideals is eventually constant), then proved that unique factorisation holds if and only if all irreducible elements … Continue reading
Posted in Notes
Tagged commutative rings, euclidean domains, irreducibles, prime ideals, primes, principal ideal domains, ring theory, rings, UFDs, unique factorisation
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Topics in Commutative Rings: Unique Factorisation (1)
Unique Factorisation: Basics Throughout this post, let R be an integral domain; recall that this means R is a commutative ring such that whenever ab=0, either a=0 or b=0. The simplest example of an integral domain is Z, the ring of integers. What’s of interest to … Continue reading
Posted in Notes
Tagged commutative rings, irreducibles, prime ideals, primes, ring theory, rings, UFDs, unique factorisation
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Introduction to Ring Theory (8)
Matrix Rings In this post, we’ll be entering the matrix. Let R be a ring. The ring Mn×n(R) is the set of matrices whose entries are elements of R, where the addition and multiplication operations are given by the usual matrix addition … Continue reading
Posted in Notes
Tagged advanced, cramer's rule, determinants, matrix rings, ring theory, rings, simple rings
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Introduction to Ring Theory (7)
Polynomial Rings A polynomial over a ring R is an expression of the form: , where , and . Let’s get some standard terminology out of the way. The element ai is called the coefficient of xi. The largest n for which an ≠ 0 is called … Continue reading
Posted in Notes
Tagged cryptography, derivatives, factor theorem, polynomials, remainder theorem, ring theory, secret sharing
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Introduction to Ring Theory (6)
Let’s keep stock of what we’ve covered so far for ring theory, and compare it to the case of groups. There are loads of parallels between the two cases. G is a group R is a ring. Abelian groups. Commutative … Continue reading
Introduction to Ring Theory (5)
Our first order of the day is to state the correspondence between the ideals and subrings of R/I and those of R. This is totally analogous to the case of groups. Theorem. Let I be an ideal of R. There are 1-1 … Continue reading
Posted in Notes
Tagged advanced, chinese remainder theorem, ideals, maximal ideals, prime ideals, ring theory, rings
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Introduction to Ring Theory (3)
Ideals and Ring Quotients Suppose I is a subgroup of (R, +). Since + is abelian, I is automatically a normal subgroup and we get the group quotient (R/I, +). One asks when we can define the product operation on R/I. To be specific, each … Continue reading
Posted in Notes
Tagged advanced, gaussian integers, generated ideals, ideals, principal ideals, ring quotients, ring theory
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Introduction to Ring Theory (2)
Subrings Just like groups have subgroups, we have: Definition. A subset S of a ring R is a subring if it satisfies the following: ; ; . The first two conditions imply that S is a subgroup of (R, +). Together with … Continue reading
Introduction to Ring Theory (1)
Recall that in groups, one has only a binary operation *. Rings are algebraic structures with addition and multiplication operations – and consistency is ensured by the distributive property. Definition. A ring R is a set together with two binary operations: … Continue reading
Posted in Notes
Tagged advanced, characteristic, commutative rings, distributive property, division rings, fields, integral domains, quaternions, ring theory, rings, units, zero-divisors
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Mathematics and Such 