The Ring Theory of Abundance
A ring is any round, typically precious metal, usually of gold, that is worn as ornamental jewelry. The word “ring” itself, however, always means jewelry worn on the hand; other pieces are designed for other purposes, e.g. rings for the wedding and engagement rings. A ring can be a single band or multi-band. The word “ring,” in the singular form, always means either jewelry worn around the finger (ring), or worn on another body part (bracelets, earrings).
An example of this type of ring would be a “belly ring.” A belly ring consists of three bands: a thick band at the base, thin band near the top that fastens to the top, and thin band at the bottom that fits around the end of the belly button. When these bands are lined up vertically, they form a perfect circle; when they are lined up horizontally (like a traditional nose ring), they form a perfect circle with each of the bands perpendicular to the rest. A horizontal line that cuts through all four of these bands represents the points of an ellipse, which can be used in mathematics to represent the algebraic structure of the ring. Therefore, when the angles between the four points of the circle are changed from right to left, the resulting angle changes the value of the corresponding value of the matrix.
To fully grasp the power of ring theory, it’s helpful to understand what happens when multiplication takes place. Each of the four subrings has an associated factor that changes when the respective factor changes. This is done in order to alter the arithmetical values of each string, so that the resulting pattern has different arithmetical values for each of the respective factors.
Multiplication can take place, for example, when I, j and k are added to each other: the product becomes bigger. Similarly, addition of numbers I, j and k to any of the four strings of a distributive ring will modify it in order to yield a different pattern. For instance, when I and k are added to any of the four sides of a right-handed ring (a b, c e), the resulting ring is called a right-handed ring, because it is oriented to read from left to right. Similarly, addition of numbers i, b, c, and d to any of the four sides of a left-handed ring (a, b, c e) makes the resulting ring a left-handed ring.
Another important concept that is needed to appreciate the connection between distributive and commutative ring theories is what multiplication entails. Commutative rings are formed when adding two values on top of one another. Any ring, in general, is a commutative ring if and only if it can be divided into smaller strings, such as a b, c, d…Where each of those smaller rings contains at least one more number.
In the previous example, the four elements of a commutative ring were I, j, k, and c, hence we know that I + j + k = zero ring. Any ring with zero elements can be easily reduced into its complement, i.e., a b, c, etc. So rings whose combinations depend on the numbers I, j, k, and c are also called zero-ring combinations. A Commutative i.e. commutative ring has no further component other than zero, so the components are easy to recognize.