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In the realm of abstract algebra, a ring stands as one fundamental algebraic structure used to generalize arithmetic operations. This consists of a set paired with two binary operations - addition and multiplication. By extending these operations, theories from arithmetic are brought into non-numerical domains like polynomials, matrices, and functions. A ring possesses an abelian group through a second binary operation that is associative, distributive over the abelian group's operation, and features an identity element. Based on the extension from integers, the abelian group's operation is addition, while the other operation is multiplication. The order of operations in a commutative ring determines whether or not elements remain unchanged under certain conditions - it does change. The noncommutativity between rings greatly impacts how these structures behave in various mathematical fields such as algebraic geometry and number theory. A crucial component in mathematics is knowing that some properties, including associativity and distributivity, exist inherently within commutative rings, thus creating new ways for theorems to be understood. Commutative ring theory or 'commutative algebra', serves a vital purpose and has undergone a profound impact from problems that originated in number theory and algebraic geometry. Some of the most basic types of commutative rings include the integer sets along with addition and multiplication, as well as polynomial additions and multiplications. These operate in affine spaces using coordinates while others such as matrix groups can serve non-commutative roles in topological spaces through group functions. The creation of ring theories emerged throughout various time frames starting from 1870. This was achieved by numerous key contributors including Dedekind, Fraenkel, Noether and most importantly David Hilbert whose efforts provided for further extension beyond number theory as well.