What is a Ring in Abstract Algebra?

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A ring is a group with additional features in abstract algebra. But first let’s look at the set of integers, set of real numbers, A matrix with 2 rows and 3 columns and the set of complex numbers.



If we look at the set of integers(Z), it is closed under addition, subtraction and multiplication meaning that if we perform these operations on two integers, we will get another integer. However, if we divide two integers, we might not get another integer but rather a fraction.

In a similar vein, the set of real numbers(R) will be closed on all these four operations provided we do not divide by 0.

In the case of R2x3 matrix, while it will be closed under addition and subtraction, we cannot multiply two matrices of arbitrary sizes but rather follow the rule and inverse of a matrix does not always exist. 

Lastly, if we take two polynomials and divide them, we get a rational function so the set of complex numbers is not closed under division.

We already have a word for a set where you can add any two elements, every element has an additive inverse, and you have the required rules from arithmetic: we call this a group. 

 Also remember from our blog on groups that in abstract algebra, we can combine two of the operations subtraction  into addition and division into multiplication.

Group

x,y in set → x+y in set

Each x has additive inverse: -x

“Rules” from arithmetic:

A+(b+c)=(a+b)+c

x+0=x, 0+y=y,…

Z,R,R2×3, C[x] are groups under addition and are also commutative under addition.

However, we have additional features to consider.

Z,R,C[x] have multiplication

R has × and multiplicative inverses

When a commutative group under addition also has multiplication, we call it a ring.

Lets now look at the textbook definition of a ring.

A ring is a set R with two operations: + . (Addition and multiplication)

Both operations are closed

If x,y ϵ R, then x+y ϵ R and x.y ϵ R


Distributive Properties:  a.(b+c) =a.b+a.c   (a+b).c=a.c+b.c

There are additional names for rings depending on how close it comes to  being a group under multiplication.

For example, while all rings are abelian under +, they may not  be commutative under ×.

R= R2×2 is not commutative


If R is commutative…

We call it a “commutative ring”

Rings may not have multiplicative inverses

Rings with inverses for × are called “division rings”

Division def is x÷y =x ×y -1

We call, Non-commutative division ring example being Quaternions.

When we have a commutative Division Ring, we call it a Field.

There you have it! Hope you got some insights of what rings are in Abstract algebra. 

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