Commutative Rings

A ring is an elegant round band, typically of metal, worn primarily as decorative jewellery. The word “ring” on its own always refers to jewellery worn on only one finger; when worn in other areas, the location is specified in the term, e.g. earrings, finger rings, nose rings, wrist rings, ankle rings, toe rings and tongue rings.


In a general sense, rings can be described in two binary operations: addition and subtraction. Any ring with one finger can be broken into two pieces, for instance a thong or a bar. Similarly, any ring can be multiplied or divided into two parts, for instance a half ring and a full ring. It follows that all rings must have a binary operation to operate on. The rings can also be described in terms of their geometry.

The idea of rings is very useful for understanding the algebraic structure of a ring. The simplest rings are those which consist of just a band and a curl, where the curl is a constant factor in the operation. The main point is that the multiplication or division is symmetrical (which is why multiplication is sometimes called “the science of numbers”). This is because the operation can be viewed as a graph of the curve on the band, with the x value on the vertical axis of the graph and the y value on the horizontal axis. The equation is then the sum of the two slopes, which defines a unique function for the ring’s variables (e.g. x and y).

Any ring can be studied algebraically as a system of pairs, where each pair is independent of every other except for the identity component which is true for every single instance of the transformation. In this way, it can be shown that the variables are uniquely defined by the functions they represent. A classic example is when we take the first term of the first quadratic equation (i.e. a b, c e), and find that if we leave out the zero i.e. x, then the second term (i.e. b) does not depend on the first term (a), but only on its values in the interval a b, c, where “a” and “b” are the constant values between the x-intercept of the first term with the x-intercept of the second term.

The most general situation of a commutative ring (in which the operation has no singular values) is when there are constant values along a curve. Here we obtain a system of definite geometrical forms. A prime example is the ring formed by the equator and the meridian. The second axis is always parallel to the first, which means that, for a fixed y, the second axis can be graphed as a curve. This leads to a great deal of the usefulness of the operation.

Another popular example of a commutative ring operation is the one obtained from the natural logarithm. Here, we have the operation called the zero-ring, meaning that there are no prime numbers involved in the operation. The zero-ring is thus symmetrical. In other words, if we cut the circle formed by the base of the earth along its equator and placed the circle inside the circle of the earth, then the circle would perfectly describe a circle having one end closer to the earth’s center than the other. The ring called the zero ring is therefore symmetrical.