Ring Theory

J(R/J)

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Field discriminant

A division ring in which the multiplication is commutative

Ring homomorphism/isomorphism

A mapping f of a ring R into a ring S is said to be a (ring) homomorphism if f(x+y)=f(x)+f(y) and f(xy)=f(x)f(y) for all x,y in R. f is an isomorphism if it is also bijective

Noetherian Module

A module with ACC on submodules is called noetherian

Artinian Module

A module with DCC on submodules is called artinian

Noether theorem

A noetherian ring has primary decomposition

Partially ordered set

A non empty set S is called partially ordered if there exists a binary operation < in S which is defined for certain pairs of elements in S and satisfies: *a <a * a< b, b<c means a<c *a<b, b<a means a=b

Rank of a prime ideal

A prime ideal P is said to have rank r if there exists a chain of prime ideas P1 prop subset P2 prop subset ... prop subset Pr prop subset P but none longer.

Division ring

A ring D with at least two elements is called a division ring if D has an identity and every non zero element of D has an inverse in D.

Integral domain

A ring R is called an integral domain if the product of any two non-zero elements in R is non-zero

Commutative ring

A ring R is commutative if ab = ba for all a,b in R

Right artinian ring

A ring with 1 and DCC on right ideals is called a right artinian ring. (Similarly for left artinian)

Right noetherian ring

A ring with ACC on right ideals is called a right noetherian ring. (Similarly for left noetherian)

Ideal, left ideal, right ideal

A subset I of a ring R is called an ideal if: *I is a subring of R *For all a in I, r in R, ar is in I and ra is in I (Right ideal only has ar in I and left ideal only has ra in I)

Subring

A subset S of R is called a subring of R if S is itself a ring wrt the laws of composition on R

Right quasi regular (rqr) set

A subset S of R is called rqr if every element of S is rqr

Meet-irreducible ideal

An ideal I is called meet-irreducible if I = A cap B, (A,B ideals of R) implies I = A or A=B

Prime ideal of a ring R

An ideal P of a ring R i said to be a prime ideal if AB subset P where A, B ideals of R implies that A subset P or B subset P. We exclude R from the list of prime ideals.

Primary ideal

An ideal Q is said to be primary if ab in Q (a,b in R) means that a is in Q or b^{n} is in Q for some integer n

The Well Ordering Principle

Any non-empty set can be well ordered. ie. it is totally ordered and every non-empty subset of it has a minimal element

Principle right (left) ideals

Cyclic submodules of R_{R} (or {R}_R)

Localisations of R at S (where S is a multiplicatively closed subset of R) We always assume 0 isn't in S and 1 is in S

Define an equivalence relation ~ on R x S as follows: (a,s) ~ (b,t) iff there exists s' in S such that (at-bs)s' = 0 Let a/s be the equivalence class of (a,s) and let R_{S} denote the set of all such equivalence classes with addition and multiplication as standard for fractions. R_{S} is called a localization of R at S

Let S be a non-empty collection of submodules of a right R module M. We say M has the ascending chain condition (ACC) for submodules in S if

Every chain of submodules A1 subset A2 subset ... with Ai in S has equal terms after a finite number of terms

Let S be a non-empty collection of submodules of a right R module M. M is said to have the maximum condition on submodules in S if

Every non-empty collection of submodules in S has a submodule maximal in this collection

Localisation of M at S (M R module, S multiplicatively closed subset of R)

FILL ME IN

Finitely generated R-modules

Generated by a finite set. If R has 1 and M is a finitely generated module then there exists a_1,...,a_n in M such that M = a_1R+ ... + a_nR

Axiom of choice

Given a class of sets, there exists a choice function. ie. a function which assigns to each of these sets one of its elements

Let S be a non-empty subset of a ring R. S is said to be nil if

Given any s in S there is an integer k such that s^{k} = 0.

Totally ordered set

If S is a partially ordered set then a non-empty subset T is said to be totally ordered if for every pair a,b in T we either have a<b or b<a

Zorn's lemma

If a partially ordered set has the property that every totally ordered subset of S has an upper bound in S, then S contains a maximal element

Product of subset

KS = {sum k_is_i where k_i in K, s_i in S} = all finite sums of elements of the type ks with k in K and s in S

Kernel/Image

Ker(f) = {x in R : f(x) = 0) Im(f) = {y in S : there is some x with f(x)=y}

Dedekind Modular Law

Let A, B, C be submodules of M_{R} such that B is a subset of A. Then A cap (B + C) = B+ (A cap C)

Normal primary decomposition

Let I = Q1 cap ... cap Qn be a primary decomposition for an ideal I. Suppose that the Qi are Pi primary. Then we say the decomposition is normal if: *No Qi is redundant *Pi not equal Pj for i not equal j

Second isomorphism theorem for rings

Let I be an ideal and L be a subring of R. Then L/(L cap I) is isomorphic to (L+I)/I

Coset

Let I be an ideal of a ring R and x be in R. Then the set of elements {x+i : i in I} is called the coset of x in R wrt I. It is denoted x+I

Third isomorphism theorem for rings

Let I, K be ideals of a ring R such that I is a subset of K. Then (R/I) / (K/I) is isomorphic to (R/K)

Right R-Submodule

Let K be a subset of a right R-module M. Then K is called a right R-submodule if K is also a right R-module under the laws of composition on M

R-homomorphism of modules, isomorphism

Let M and M' be right R-Modules. A mapping f: M to M' is called an R-homomorphism if: *f(x+y) = f(x) + f(y) for all x,y in M *f(xr) = f(x)r for all x in M, r in R f is an isomorphism if it is also bijective

Isomorphism theorems for modules

Let M and M' be right R-modules and o : M to M' an R-homomorphism. Then o(M) isomorphic to M/K where K = ker(o) Let L, K be submodules of M_R. Then (L+K)/K isomorphic to L/(L cap K) If K, L are submodules of M_R and K subset L then L/K is a submodule of M/K and (M/K)/(L/K) isomorphic to M/L

Maximal right ideal

Let M be a right ideal of R. M is said to be a maximal right ideal if M isn't equal to R and M proper subset of M' where M' is a right ideal of R must meant that M' = R

Radical of a primary ideal, P-Primary ideal

Let Q be a primary ideal. Let P/Q be the nilpotent radical of the ring R/Q. P is called the radical of and we say that Q is P-Primary.

Local ring

Let R be a commutative ring. R is said to be a local ring if R has a unique maximal ideal

Ring

Let R be a non-empty set which has two laws of compositions definted on it We say that R is a ring if it an abelian group under addition and: * ab is in R for all a,b in R * a(bc) = (ab)c for all a,b,c in R * a(b+c) = ab+ac and (a+b)c = ac +bc for all a,b,c in R

Nakayama's lemma

Let R be a ring with 1 and M_{R} a fg modules. Then MJ = M means M=0

Unital module

Let R be a ring with 1. A module M_{R} is said to be unital if m1=m for all m in M.

Right R-Module

Let R be a ring. A set M is called a right R-Module if: *M is an abelian group *A law of composition M x R to M is defined *(x+y)r = xr+yr *x(r+s) = xr+xs *x(rs) = (xr)s

External direct sum of rings

Let R_1, ...,R_n} be rings. We define the external direct sum S to be the set of all n-tuples {(r_1,...,r_n) : r_i in R_i}. We define addition and multiplication component-wise.

Multiplicatively closed subset

Let S be a non-empty subset of a ring R. We say that S is multiplicatively closed if s,t inn S implies st is in S

Maximal element (wrt partially ordered sets)

Let S be a partially ordered set. An element s in S is a maximal element if x< y with y in S must mean x=y

Upper bound (wrt partially ordered sets)

Let T be a totally ordered subset of a partially ordered set S. We say that T has an upper bound in S if there is a c in S such that x < c for all x in T

Prime P minimal over a1,...,an

Let a1,...,an be in R. We say that prime P is minimal over a1,...,an if P/(a1R+...+anR) is a minimal prime of the ring R/(a1R+...+anR)

Regular element

Let c be in R. We say that c is regular if cx = 0 , x in R implies x=0. An element which is not-regular is a zero-divisor

First isomorphism theorem for rings

Let f be a homomorphism of a ring R onto a rings S. Then f(R) is isomorphic to R/I where I is ker(f)

Right quasi regular (rqr) element

Let x be an element of a ring R. Then we say that x is rqr if 1-x has a right inverse. ie. if there is a y in R such that (1-x)y=1

Internal direct sum of rings

Let {I_{l}} be a collection of ideals of a ring R. We define their sum to be sum_{l} I_{l} = {x = x_1 +x_2+ ... +x_{k} : x_i in I_l_i for k = 1,2,3,...} ie. the collection of finite sums of the elements of the I_l The sum is direct if each element of the sum is uniquely expressible as a sum.

Cyclic modules

M_{R}R generated by a single element.

P prime ideal, nth symbolic power of P

P^{(n)} = {x in R st xc is in P^{n} for some c in C(P)}

Primary ring

R is called a primary ring if 0 is a primary ideal

Residue class ring

R/I = set of all ideals x+I is the residue class ring of R wrt I

Right ideals defined by submodules

Submodules of R_{R} are called right ideals of R and submodules of {R}_R are called left ideals of R

Jacobson Radical

The intersection of all maximal right ideals of a ring R is called its Jacobson Radical. It is usually denoted by J(R) or just J

External direct sum of modules

The set of n-tuples {(m_1,...,m_n): m_i in M_i}

Let T be a subset of M_{R}. What is the submodule of M generated by T

The smallest submodule of M containing T. ie. The intersection of all submodules of M containing T By convention we take {0} to be the submodule generated by the empty set.

Nilpotent radical

The sum of all nilpotent right ideals of R is called the nilpotent radical and is denoted by N(R)

Let S be a non-empty subset of a ring R. S is said to be nilpotent if

There exists an integer k such that S^{k} = 0

Quasi regular (qr) element or set

We say an element (or set) is quasi regular if it is both rqr and lqr

Primary decomposition

We say that R has primary decomposition if every ideal of R is expressible as a finite intersection of primary ideals

Minimal prime ideal of R

We say that a prime ideal P is a minimal prime ideal of R if Q subset P with Q prime implies Q = P

Let I be a right ideal of R. We say that a1, a2,...,an is a minimal generated set for I if

a1,...,an generate I no proper subset of a1,...,an generates I

Annihilator of S in R (S subset of R)

ann(S) = {r in R : Sr=0}

Natural homomorphism of R onto R/I

c : R to R/I defined by c(x) = x+I (where I is an ideal of R)

V(R)

dim J/J^2 as a vector space over the field R/J

Let S be a non-empty collection of submodules of a right R module M. We say an element K in S is maximal if

there is no K' in S such that K is a proper subset of K'

Unit of a ring

u in R is a unit if there exists v in R with uv = vu = 1

Submodule generated by a

{ar + la : r in R, l in Z} This equals aR when R has 1 and M is unital

Let I be an ideal of R. C(I) =

{x in R : x+I is regular in the ring R/I}