CARRRIER LOVE 70 3D print model

cgtrader

His article is about an algebraic structure known as a ring. For geometric structures related to this concept, see Annulus (mathematics). In set theory, it refers to the Ring of sets. Chapter IX of David Hilbert's book on number theory, Die Theorie der algebraischen Zahlkorper, defines and discusses the properties of a number field. The chapter is titled "Die Zahlringe des Korpers", which translates to "The Number Rings of the Field". The term "ring" comes from the contraction of "Zahlring". A ring is one of the basic algebraic structures in mathematics that underlies abstract algebra. It consists of a set equipped with two binary operations that generalize arithmetic addition and multiplication. By applying this generalized framework, theorems developed for numbers are extended to non-numerical objects like polynomials, series, matrices, and functions. A ring can be viewed as an abelian group equipped with another binary operation. This additional operation is associative, distributed over the group's operation, and features an identity element. Although some authors do not require this last property, as noted in the definition section. Since rings generalize integers, we denote the abelian group operation as addition and the second binary operation as multiplication. The question of whether a ring is commutative or noncommutative has significant implications for its behavior as an abstract object. Commutative ring theory, also known as commutative algebra, is a crucial aspect of ring theory that studies the properties and behavior of commutative rings. This area of study has been greatly influenced by problems arising from algebraic number theory and algebraic geometry. Some examples of commutative rings include the set of integers with standard addition and multiplication operations, the set of polynomials with their corresponding addition and multiplication, and the ring of integers in a number field. In contrast, noncommutative rings feature multiplication operations that depend on the order in which elements are multiplied. Examples of noncommutative rings include n by n real square matrices (n greater than or equal to 2), group rings used in representation theory, operator algebras found in functional analysis, and cohomology rings in topology. The concept of a ring was developed in the late 1800s and further formalized by mathematicians like Dedekind, Hilbert, Fraenkel, and Noether. Initially motivated by their work on number theory, algebraic geometry, and invariant theory, they generalized certain structures that had arisen from those areas into what we now call a ring. As mathematical theories expanded beyond their initial domains, the study of rings found applications in new fields such as geometry and analysis. Today, understanding and classifying various types of rings is a key area of research in abstract algebra and related branches of mathematics.