Zero

A (real or complex) number having the property that any number does not change if zero is added to it. It is denoted by the symbol $0$. The product of any number with zero is zero: $$0\cdot a = a \cdot 0 = 0 .$$

If the product of two numbers is zero, then one of them is zero (that is, $a\cdot b = 0$ implies $a=0$ or $b=0$). Division by zero is not defined.

A direct generalization of this concept is that of the identity element of an Abelian group: the zero of an Abelian group $A$ (in additive notation) is an element, also denoted by $0$, satisfying $0+a = a$ for all $a \in A$. It is uniquely determined.

The zero of a ring (in particular, of a skew-field, i.e. division ring, or a field) is the zero of its additive group. The zero of a ring (like the number $0$) is an absorbing element for multiplication: $0\cdot a = a \cdot 0 = 0$. However, in an arbitrary ring the product of two non-zero elements may be zero. Such elements are called zero divisors. Fields, skew-fields and integral domains do not have zero divisors.

A left zero of a semi-group $A$ (in multiplicative notation) is a left absorbing element $0 \in A$ such that $0\cdot a = 0$ for all $a \in A$. A right zero is defined by the dual property. If a semi-group has a two-sided zero (an element which is both a left and a right zero), then this element is unique. The zero of a ring is also the zero of its multiplicative semi-group.

The zero of a partially ordered set, such as a lattice, is a minimal element, if this exists. A complete lattice always has a zero: the intersection of all elements.

A zero of an algebraic system is an element picked out by a nullary operation (see Algebraic operation; Algebraic system). In the majority of examples considered above the zero is unique in the given system and even forms a one-element subsystem.

A zero is also called a null element.

For a zero object of a category, see Null object of a category.

The set of zeros of a function $f(x_1,\ldots,x_n)$ taking values in an Abelian group (ring, field, skew-field) $A$ is the collection of values of the variables $(x_1,\ldots,x_n)$ for which $f(x_1,\ldots,x_n) = 0$.

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A subset of a topological space $X$ is called a zero set if it is the set of zeros of some continuous real-valued function on $X$. Zero sets are an object of study in algebraic geometry (zero sets of systems of polynomials) and local analytic geometry (zero sets of systems of holomorphic functions and mappings).

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