Algebra
Abstract algebra

Abstract algebra, also called modern algebra, studies different types of algebraic structures. An algebraic structure is a framework for understanding operations on mathematical objects, like the addition of numbers. While elementary algebra and linear algebra work within the confines of particular algebraic structures, abstract algebra takes a more general approach that compares how algebraic structures differ from each other and what types of algebraic structures there are, such as groups, rings, and fields. The key difference between these types of algebraic structures lies in the number of operations they use and the laws they obey.

On a formal level, an algebraic structure is a set of mathematical objects, called the underlying set, together with one or several operations. Abstract algebra is primarily interested in binary operations, which take any two objects from the underlying set as inputs and map them to another object from this set as output. For example, the algebraic structure ⟨ N, + ⟩ {\displaystyle \langle \mathbb {N},+\rangle } has the natural numbers (N {\displaystyle \mathbb {N} }) as the underlying set and addition (+ {\displaystyle +}) as its binary operation. The underlying set can contain mathematical objects other than numbers and the operations are not restricted to regular arithmetic operations. For instance, the underlying set of the symmetry group of a geometric object is made up of geometric transformations, such as rotations, under which the object remains unchanged. Its binary operation is function composition, which takes two transformations as input and has the transformation resulting from applying the first transformation followed by the second as its output.

Abstract algebra classifies algebraic structures based on the laws or axioms that its operations obey and the number of operations it uses. One of the most basic types is a group, which has one operation and requires that this operation is associative and has an identity element and inverse elements. An operation is associative if the order of several applications does not matter, i.e., if (a ∘ b) ∘ c {\displaystyle (a\circ b)\circ c} is the same as a ∘ (b ∘ c) {\displaystyle a\circ (b\circ c)} for all elements. An operation has an identity element or a neutral element if one element e exists that does not change the value of any other element, i.e., if a ∘ e = e ∘ a = a {\displaystyle a\circ e=e\circ a=a}. An operation has inverse elements if for any element a {\displaystyle a} there exists a reciprocal element a − 1 {\displaystyle a^{-1}} that undoes a {\displaystyle a}. If an element operates on its inverse then the result is the neutral element e, expressed formally as a ∘ a − 1 = a − 1 ∘ a = e {\displaystyle a\circ a^{-1}=a^{-1}\circ a=e}. Every algebraic structure that fulfills these requirements is a group. For example, ⟨ Z, + ⟩ {\displaystyle \langle \mathbb {Z},+\rangle } is a group formed by the set of integers together with the operation of addition. The neutral element is 0 and the inverse element of any number a {\displaystyle a} is − a {\displaystyle -a}. The natural numbers with addition, by contrast, do not form a group since they contain only positive integers and therefore lack inverse elements. Group theory is the subdiscipline of abstract algebra which studies groups.

A ring is an algebraic structure with two operations (∘ {\displaystyle \circ } and ⋆ {\displaystyle \star }) that work similarly to addition and multiplication. All the requirements of groups also apply to the first operation: it is associative and has an identity element and inverse elements. Additionally, it is commutative, meaning that a ∘ b = b ∘ a {\displaystyle a\circ b=b\circ a} is true for all elements. The axiom of distributivity governs how the two operations interact with each other. It states that a ⋆ (b ∘ c) = (a ⋆ b) ∘ (a ⋆ c) {\displaystyle a\star (b\circ c)=(a\star b)\circ (a\star c)} and (b ∘ c) ⋆ a = (b ⋆ a) ∘ (c ⋆ a) {\displaystyle (b\circ c)\star a=(b\star a)\circ (c\star a)}. The ring of integers is the ring denoted by ⟨ Z, +, × ⟩ {\displaystyle \langle \mathbb {Z},+,\times \rangle }. A ring becomes a field if both operations follow the axioms of associativity, commutativity, and distributivity and if both operations have an identity element and inverse elements. The ring of integers does not form a field because it lacks multiplicative inverses. For example, the multiplicative inverse of 7 {\displaystyle 7} is 1 7 {\displaystyle {\tfrac {1}{7}}}, which is not part of the integers. The rational numbers, the real numbers, and the complex numbers each form a field with the operations addition and multiplication.

Besides groups, rings, and fields, there are many other algebraic structures studied by abstract algebra. They include magmas, semigroups, monoids, abelian groups, commutative rings, modules, lattices, vector spaces, and algebras over a field. They differ from each other in regard to the types of objects they describe and the requirements that their operations fulfill. Many are related to each other in that a basic structure can be turned into a more advanced structure by adding additional requirements. For example, a magma becomes a semigroup if its operation is associative.
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