In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton...

of vector spaces over a field. See Zero object (algebra) for details. This is the origin of the term "zero object". In Ring, the category of rings with...

of rings, the zero ring is the terminal object, whereas the ring of integers Z is the initial object. The zero ring, denoted {0} or simply 0, consists...

In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may...

"tangible" mathematical objects. A spectral sequence is a powerful tool for this. It has played an enormous role in algebraic topology. Its influence...

the trivial non-strict order ⁠ ≤ {\displaystyle \,\leq } ⁠. Zero object (algebra) – Algebraic structure with only one element List of small groups Rowland...

is zero Zero (complex analysis), a zero of a holomorphic function Zero element, generalization of the number zero in algebraic structures Zero object (algebra)...

Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper...

functions on X. The function field of an algebraic variety X (a geometric object defined as the common zeros of polynomial equations) consists of ratios...

and classification of Lie groups in terms of Lie algebras, which are simpler objects of linear algebra. In more detail: for any Lie group, the multiplication...

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as...

numbers R {\displaystyle \mathbb {R} } and all algebraic number fields. Other fields of characteristic zero are the p-adic fields that are widely used in...

algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects...

number of objects is even. If an object is left over, then the number of objects is odd. The empty set contains zero groups of two, and no object is left...

the object of study—this is the subject of group theory and ring theory— in universal algebra, the object of study is the possible types of algebraic structures...

In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called addition and multiplication, which obey the same...

algebraic geometry: if R is a ring of functions defined on some geometric object (algebraic variety) V, and one wants to study this variety "locally" near a point...

The action of a monoid (= commutative ring) R on an object (= ring) A of Ring is an R-algebra. The category of rings has a number of important subcategories...

Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b...

numbers, as well as other algebraic structures. Multiplying any number by 0 results in 0, and consequently division by zero has no meaning in arithmetic...

set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathematical objects together with one or several operations...

category of R-algebras to the category of sets. Free algebras over division rings are free ideal rings. Cofree coalgebra Tensor algebra Free object Noncommutative...

unital associative R-algebra is a monoid object in R-Mod (the monoidal category of R-modules). By definition, a ring is a monoid object in the category of...

universal algebra, an algebraic structure is called an algebra; this term may be ambiguous, since, in other contexts, an algebra is an algebraic structure...

Kernels allow defining quotient objects (also called quotient algebras in universal algebra). For many types of algebraic structure, the fundamental theorem...

numbers, any algebraic expression can be called an arithmetic expression. However, algebraic expressions can be used on more abstract objects such as in...

generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the...

F(A)\rightarrow A} . The object A {\displaystyle A} is called the carrier of the algebra. When it is permissible from context, algebras are often referred to...

In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. This initiality provides a general framework...

every element of L is a root of a non-zero polynomial with coefficients in K. A field extension that is not algebraic, is said to be transcendental, and...