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The chapter is titled Die Zahlringe des Korpers, literally the number rings of the field. This term comes from Zahlring, a contraction of the word ring in German. A ring in mathematics refers to an abstract algebraic structure. It consists of a set that contains two binary operations. These operations are a generalization of the addition and multiplication seen in arithmetic. Through this process, many of the arithmetic theorems are extended to non-numerical objects like matrices, polynomials, series, and functions. A ring must be an abelian group. This means it has a second operation that is associative and distributes over the first group operation. Additionally, a ring typically contains an identity element for its multiplication. However, not all definitions of a ring include this last requirement. Some definitions also use the generalization of the integers' properties to label these two operations. A question arises whether a ring follows the commutative rule or not. The result often has a major impact on the behavior of an abstract object in mathematical equations and formulas. In turn, researchers who focus on such mathematical questions and ideas naturally gravitate towards problems from both algebraic number theory and algebraic geometry. Rings can take different forms. These include integers equipped with addition and multiplication, polynomials with their respective operations, the ring of a polynomial, coordinate rings for an algebraic variety, and finally integer numbers of fields in various mathematical areas. As opposed to this list are rings which aren't considered as being commutative; matrices which can change with each value in them but follow no certain form except when there are only 2 matrices together and one group ring, a kind used for operators which act on sets called spaces to transform or modify their internal state based on rules. Another ring that doesn't fall under this list is that of operators used for functions known as algebras from functional analysis. Then you also have a certain topological space within the cohomology ring that serves many applications both inside mathematics and outside in real-world science