Abstract Nonsense

Crushing one theorem at a time

Basic Definitions of Rings


Point of Post: In this post we begin our discussion of rings with definitions and examples.

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Motivation

The notion of a group started out simply. Namely, we wanted to describe some kind of set with a function where the elements interact in a respectable way. Namely, we wanted them to interact associatively, have an identity element, and each element to possess an inverse. The natural question after studying groups is “what about sets with two operations?” Thus, one begins to seek out a definition of a set with two binary operations that is as fruitful as that of a group. So the first obvious thing one asks is what does one precisely mean by having “two operations”? It is very easy to take a set G and endow it with two different group structures, is this what we mean? Clearly not since the study of these two operations really just reduces to the study of one operation–just doing it twice. Thus, to make the notion of ‘two operations’ more rich than just two disparate operations what we would like for the two operations themselves to interact. After much experimentation mathematicians (taking natural examples such as the endomorphism algebra with its two usual operations and \mathbb{Z} with its two usual operations) decided upon the object which is now known as a ring. Roughly a ring is an abelian group where we have some notion of ‘multiplying’ (just a name for the second operation besides addition) the group elements in a way which is associative and which distributes over the addition. That’s it. There is no requirement that the multiplication be commutative or have some form of identity element. Both of these are common addons to rings, but in general aren’t in the definition. Whenever one deals with rings, the thing to keep in mind are the integers–they are the measuring stick by which most ring-theoretic concepts are easily measured.

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Definitions 

An ordered triple \left(R,+,\cdot\right) is called a ring if (R,+) is an abelian group and \cdot:R\times R\to R (in practice we often just use concatenation to denote multiplication) is an associative binary operation such that a\cdot(b+c)=a\cdot b+a\cdot c and (b+c)\cdot a=b\cdot a+c\cdot a for all a,b,c\in R (this is known as distributivity). If \cdot is a commutative binary operation we call R a commutative ring. If there exists 1\in R such that 1\cdot r=r\cdot 1=r for all r\in R we say that R is unital or that R has 1. This 1, if it exists, is called the multiplicative identity for R and it’s clear that it is unique (this follows since (R,\cdot) is a monoid). We will often implicitly assume that 1\ne 0, although this not be true, but it is clear that if 1=0 then R=\{0\}.

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There is a long list of easily verifiable facts about rings such as 0\cdot a=0, (-a)\cdot(-b)=a\cdot b, -a=-1\cdot a, etc.

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If R is unital and u\in R is such that there exists v\in R with uv=vu=1 we say that u is a unit. We denote the set of all units by R^\times and note that this is trivially a group. When we want to think of R^\times as a (multiplicative) group we call it the group of units and denote it U(R).  A unital ring R is called a division ring if R^\times=R-\{0\}. A field is a commutative division ring.

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A zero divisor in a ring R is an element r such that there exists s\in R with either rs=0 or sr=0. It’s evident that a zero divisor cannot be a unit.

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References:

Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

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June 15, 2011 - Posted by | Algebra, Ring Theory | , , ,

12 Comments »

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