Rings, Integral Domains and Fields

This is a note about abstract algebra, one of the largest branches of mathematics. I will show some concepts in abstract algebra and the power of abstraction.

What is abstraction?

Before getting our hand's dirty with some algebra, first let us consider the concept of abstraction. Abstraction is, loosely speaking, the process of taking a problem or a concept and removing unnecessary context. This is an important concept in mathematics. It is what allowed us to view the number 33 for example, as an abstract concept rather than 33 lots of sheep. Another context we could think about is shapes. Let's say we know a fact about a triangle and a square. We later do some observation to reveal that this also applies to a pentagon and hexagon and so on. However how do we prove that this is the case? Well we cannot prove it for every polygon we can come up with because there are infinitely many! But what if we proved the hypothesis for an nn-gon. Independent of the number of vertices. Then we have automatically proven it for all polygons! This is the power of abstraction.

Sets

Sets are arguably the most important concepts in Mathematics. All of Mathematics can be derived from sets! However i will not show this here. I am interested in some special sets. These sets are used on a daily basis and by explaining the structures they possess, it helps give a better understanding of the. Firstly let us look at some commonly used sets:

  • Z\Bbb Z the set of integers. {,2,1,0,1,2,}\{\cdots, -2, -1, 0, 1, 2,\cdots\}
  • Q\Bbb Q the set of rational numbers. Numbers that can be expressed as a ratio of integers q=a/bq=a/b where qq is rational and aa and bb are integers.
  • R\Bbb R the set of real numbers. This includes rational and irrational numbers. {e,π,2,3,1/2,}\{e, \pi, \sqrt{2}, 3, 1/2,\cdots\}
  • C\Bbb C the set of complex numbers. {3+4i,e+πi,6i,2,}\{3+4 i, e+\pi i, 6i, 2,\cdots\}
  • Zn\Bbb Z_n the set of integers modulo nn {0,1,2,,n1}\{0,1,2,\cdots,n-1\}

You are probably familiar with these sets, if you are not then this post will not make much sense to you. One thing to note is that all of these sets share many algebraic properties. These properties define an abstract object called a ring.

Definition 1 A ring RR is a set of elements on which two binary operations, addition (+)(+) and multiplication ()(\cdot), are defined that satisfy the following properties for all a,b,cRa,b,c\in R:

  1. (Addition is commutative) a+b=b+aa+b=b+a
  2. (Addition is associative) (a+b)+c=a+(b+c)(a+b)+c=a+(b+c)
  3. (Additive identity exists) There exists an element 00 in RR such that a+0=aa+0=a
  4. (Additive inverses exist) For each element aa in RR, there exists an element xx such that a+x=0a+x=0
  5. (Multiplication is associative) (ab)c=a(bc)(a\cdot b)\cdot c=a\cdot (b\cdot c)
  6. (Multiplication distributes over addition) a(b+c)=ab+aca\cdot (b+c)=a\cdot b+a\cdot c and (b+c)a=ba+ca (b+c)\cdot a=b\cdot a+c\cdot a

Note that multiplication in a ring is not always commutative. A ring where multiplication is commutative is called a commutative ring. An example of a non-commutative ring would be M2(Z)M_2(\Bbb Z), the collection of 2×22\times2 matrices with integer entries. Remember, matrix multiplication is not commutative.

Already we can prove some things about rings,. Suppose RR is a ring and a,bRa,b\in R then:

  1. (Additive Cancellation). If a+b=a+ca+b=a+c, then b=cb=c.
  2. (Solution of equations) The equation a+x=ba+x=b always has a unique solution in RR.
  3. (Uniqueness of additive inverses) Every element of RR has exactly one additive inverse.
  4. (Uniqueness of additive identity) There is only one element of RR that satisfies the equations z+a=az+a=a, for all aa: namely the element 00.

These properties of rings are not in the definition but derive directly from the ring definition. It is fun to prove these however remember that RR is not necessarily commutative so the order of multiplication cannot be switched.

Now let us look at more concepts about rings.

Definition 2

A subset SS of a ring RR is said to be a subring of RR if SS is itself a ring under the operations induced from RR.

This is an important concept because it tells us that ring structure can be inherited for example QR\Bbb Q \subset \Bbb R and both are rings therefore Q\Bbb Q is a subring of R\Bbb R.

The Subring theorem A non-empty subset of a ring is a subring under the same operations if and only if it is closed under multiplication and subtraction.

This is an important theorem because it allows us to check if a subset is indeed a subring without checking every property of a ring. This can be proven however it will not be proven here.

Some rings RR have a unity, or multiplicative identity; that is, an element uRu\in R where au=ua=aau=ua=a for all aRa \in R. If a unity exists, it is unique.

For example the number 11 is the unity in Z\Bbb Z, Q\Bbb Q, R\Bbb R and C\Bbb C. The matrix (1001)\left(\begin{matrix}1&0\\0&1\end{matrix}\right) is the unity in M2(ZM_2(\Bbb Z). But 2Z2\Bbb Z, the set of even integers {,4,2,0,2,4,}\{\cdots, -4, -2, 0, 2, 4,\cdots\}, has no unity.

An element a0a\ne0 is a zero divisor if there is an element b0b\ne0 with ab=0ab=0. For example in Z6,23=0\Bbb Z_6,\quad 2\cdot3=0

A commutative ring with unity is an integral domain if it has no zero divisors. Z\Bbb Z, Q\Bbb Q, R\Bbb R and C\Bbb C are all integral domains. The integers modulo nn, Zn\Bbb Z_n, is only an integral domain if and only if nn is prime.

Integral domains have the nice property of multiplicative cancellation. We can state this as a theorem:

If RR is an integral domain and a,b,cRa,b,c\in R with a0a\ne0, then ab=acab=ac implies that b=cb=c.

If RR is a ring with unity 11, then an element aRa\in R is a unit if there exists a bRb\in R such that ab=1ab=1. In this case, bb is said to be the multiplicative inverse of aa. If all the non-zero elements of a commutative ring with unity are units, then we say the ring is a field. The rings Q\Bbb Q, R\Bbb R and C\Bbb C are all fields but Z\Bbb Z is not. All fields are integral domains. Zp\Bbb Z_p is a field, for a prime pp. Indeed, all finite integral domains are fields. The fact that Zp\Bbb Z_p is a field is an important setting for proving Fermat's little theorem which states that:

If pp is prime and 0<x<p0<x<p, then xpx(modp)x^p\equiv x\pmod p.

Finally, consider F[x]F[x], the set of polynomials over an arbitrary field FF with coefficients in FF. F[x]F[x] is an integral domain. The Division Theorem, the Root Theorem, Euclid's Algorithm, the GCD identity and Unique Factorization all hold for F[x]F[x]. A particularly important example is F=CF=\Bbb C, the field of complex numbers. C[x]\Bbb C[x] satisfies the Fundamental Theorem of Algebra: Every non-constant polynomial in CC has a root.

There are a lot of theorems mentioned earlier. You may recognise some of them but as you can see the power of abstract algebra has generalised them for the field of polynomials F[x]F[x] rather than for integers.

I hope this post has enlightened you about the power of abstract algebra and given you a taste of higher Mathematics. The last paragraph about polynomials over fields is an interesting thing that has many fun proofs to try. But an interesting note to take is that all of these things that we do with integers, arithmetic, GCD and factorisation and so on can be generalised to polynomials and other sets.

Note by Ali Caglayan
5 years ago

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Very nice post, Ali! Very helpful!

Cody Johnson - 5 years ago

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Brilliant seriously needs wikis on higher pure mathematics, which I believe to contain the true flavour of mathematics. Anyways, a nice post.

Kuldeep Guha Mazumder - 3 years, 4 months ago

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yes it would be great also to have exercises about abstract algebra

guido barta - 3 years, 3 months ago

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No not only exercises, theory is of utmost necessity. Though Brilliant focusses primarily on problem solving but I being essentially a proponent of theory developing (I don't underrate problem solving by any means), think some necessary focus must be put on the theory of these topics of pure mathematics.

Kuldeep Guha Mazumder - 3 years, 3 months ago

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Nice brief intro to abstract algebraic structures

Ethan W - 4 years, 2 months ago

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