Rings are a class of objects in mathematics that generalize the integers and other common mathematical objects like polynomials, matrices, modular arithmetic and more. They are also related to fields and are closely associated with number theory.
Ring-Theoretic Concepts
In mathematics, rings are commutative groups under addition that have a second operation: multiplication. They generalize a wide variety of mathematical objects and are used in many areas of math including number theory, algebraic geometry, and invariant theory. They also play an important role in the development of modern commutative algebra.
If all nonzero elements in a commutative ring with unity have a multiplicative inverse, the ring is called a field. These ring-theoretic concepts are fundamental objects in many different branches of mathematics and are widely studied.
Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has profound implications on its behavior. These implications lead to a large number of corresponding problems and ideas in various areas of mathematics such as algebraic number theory, algebraic geometry, and commutative algebra.
A ring is often seen as a geometric representation of numbers or an extension of the integers to include more complex numbers. This can make ring-theoretic concepts easier to understand and apply in a wide range of areas of math.
The concept of a ring was first formalized in the 1870s by Dedekind and Hilbert. Rings are a generalization of Dedekind domains that occur in number theory and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory.
They are a fundamental object in number theory and algebraic geometry, and they have a wide range of applications in other areas of mathematics such as linear functions and the axioms of commutative algebra.
Their basic properties are that a ring is a set of nonzero elements and that the number of such elements in the set is finite. The simplest examples of a ring with unity are the integers, which are commutative but nonzero, and the commutative polynomials.
Some rings have multiplicative inverses, while others do not. If a commutative ring with a multiplicative inverse has more than one element, it is called a division ring or a skew field. The simplest of these types of fields are strictly commutative, but some non-commutative ones exist as well.
Rings can be made out of a variety of materials and they can be fashioned into unique designs. If you want to get your hands on the perfect ring, ask your local jeweler about custom options. They can work with you to design a unique, one-of-a-kind piece of jewelry that you and your future spouse will cherish for years.
The ring system of planets is thought to have been formed when a series of moons were thrown off course in their orbits and collided. Some of these moons stayed behind to form the ring systems, while others were destroyed and lost their ability to form new moons.