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I'm teaching myself algebra (programmer by profession), and I wanted to clear a constant doubt that I'm having.

My doubt arose from the self-study of algebraic structures- Groups, Fields and Rings and these SE hits:

What are the differences between rings, groups, and fields?

Can anyone explain the difference between a ring, a group, and a field in a way so that your average 15 year old can understand?

Let's use the partial definition of a group as:

A group is a set of elements $E$ with an operation $\circ{}$ to combine the elements of $E$....

Most of the answers above use addition, subtraction, multiplication, and division as the operation $\circ$ to explain these concepts. Is there some special reason behind this other than illustration purposes? My programmer brain wonders why can't I use any lambda with binary arity instead, of course, as long as it follows the definitional requirements of the particular structure? Say, lambda x,y: math.log(x,y).

That leads to the last related question- an integer ring $Z_m$ consists of:

  1. The set $Z_m$ = $\{0, 1, ..., m-1\}$
  2. Two operations $+$, and $\times$ such that ....

By now, I know that the inverse of operation $+$ exists, but not necessarily for $\times$ for an integer ring. Then, can I assume that there are rings out there that could be working with different sets and operations that are not even remotely close to the arithmetic operations of addition and multiplication?

DaveIdito
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    Welcome to Mathematics Stack Exchange. For some rings, composition of functions is used instead of multiplication (and may not be commutative) – J. W. Tanner Jul 17 '20 at 18:13

2 Answers2

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Ultimately, I would say that it depends on what you would like to do with your binary operation $\circ.$ Given a set $S$ with a binary operation $\circ,$ you have a magma. Going one step further, you could make sure that $\circ$ is associative, and you'd wind up with a semigroup. If you desire an identity element $e_S,$ then you are dealing with a monoid. Last but not least, requiring your elements to have inverses under $\circ$ will give you a group.

Given that $S$ is a commutative group under $\circ,$ an additional binary operation $*$ will make $S$ into a ring if

1.) $S$ is a monoid under $*$ and

2.) the operation $*$ is distributive, i.e., $r * (s \circ t) = (r * s) \circ (r * t)$ and $(s \circ t) * r = (s * r) \circ (t * r).$

Consequently, the answer to your last question is probably no because the operation $\circ$ acts in an analogous manner to addition of integers, and the operation $*$ acts similarly to multiplication of $n \times n$ matrices.

Dylan C. Beck
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It depends on what you mean by 'remotely close', but you could consider the following: Take any set $X$ (if a ring needs to have a multiplicative identity, $X$ may not be empty). Now on the power set $\mathcal P (X)$ define addition of $A,B\subseteq X$ via $A\Delta B := (A\setminus B) \cup (B\setminus A)$ and multiplication as $A\cap B$. Then $\mathcal P(X)$ is a ring.

Rings of this type are called Boolean rings, since they arise when you study Boolean algebras.

cuperius
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