- List of zero terms
The number 0 is an important concept in

mathematics .**Zero module**In

mathematics , the**zero module**is the module consisting of only the additive identity for the module'saddition function. In theinteger s, this identity is zero, which gives the name "zero module". That the zero module is in fact a module is simple to show; it is closed under addition andmultiplication trivially.**Zero ideal**In

mathematics , the**zero ideal**in a ring $R\; is\; the\; ideal\; \{\; 0\; \}$ consisting of only the additive identity (or zero element). It is immediate to show that this is an ideal.**Zero matrix**In

mathematics , particularlylinear algebra , a**zero matrix**is a matrix with all its entries being zero. Some examples of zero matrices are:$0\_\{1,1\}\; =\; egin\{bmatrix\}0\; end\{bmatrix\},\; 0\_\{2,2\}\; =\; egin\{bmatrix\}0\; 0\; \backslash 0\; 0\; end\{bmatrix\},\; 0\_\{2,3\}\; =\; egin\{bmatrix\}0\; 0\; 0\; \backslash 0\; 0\; 0\; end\{bmatrix\},$

The set of "m"×"n" matrices with entries in a ring K forms a ring $K\_\{m,n\}\; ,$. The zero matrix $0\_\{K\_\{m,n\; ,$ in $K\_\{m,n\}\; ,$ is the matrix with all entries equal to $0\_K\; ,$, where $0\_K\; ,$ is the additive identity in K.

:$0\_\{K\_\{m,n\; =\; egin\{bmatrix\}0\_K\; 0\_K\; cdots\; 0\_K\; \backslash 0\_K\; 0\_K\; cdots\; 0\_K\; \backslash vdots\; vdots\; vdots\; \backslash 0\_K\; 0\_K\; cdots\; 0\_K\; end\{bmatrix\}$

The zero matrix is the additive identity in $K\_\{m,n\}\; ,$. That is, for all $A\; in\; K\_\{m,n\}\; ,$ it satisfies

:$0\_\{K\_\{m,n+A\; =\; A\; +\; 0\_\{K\_\{m,n\; =\; A$

There is exactly one zero matrix of any given size "m"×"n" having entries in a given ring, so when the context is clear one often refers to "the" zero matrix. In general the zero element of a ring is unique and typically denoted as 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.

The zero matrix represents the

linear transformation sending all vectors to thezero vector .**Zero tensor**In

mathematics , the**zero tensor**is anytensor , of any rank, all of whose entries are zero. The zero tensor of rank 1 is sometimes known as thenull vector .Taking a

tensor product of any tensor with any zero tensor results in another zero tensor. Adding the zero tensor is equivalent to the identity operation.**Zero divisor**A

in a ring "R" is an element "a" ∈ "R" such that "ab" = 0 for some non-zero "b" ∈ "R".zero divisor **Zerosumfree monoid**In

abstract algebra , an additivemonoid $(M,\; 0,\; +)$ is said to be**zerosumfree**if nonzero elements do not sum to zero. Formally::$(forall\; a,bin\; M)\; a\; +\; b\; =\; 0\; implies\; a\; =\; 0\; !$

This means that the only way zero can be expressed as a sum is as $0\; +\; 0$.

**Zero set**In

mathematics , theof a real-valued function "f" is the inverse image $f^\{-1\}(0)$.zero set **Zero order**A

**zero order**process is one in which past behavior does not affect the future outcome. For example, inmarketing , the zero orderhypothesis holds that thebrand s you will purchase next will not depend on the brands you have purchased before. That means you have a fixedprobability of purchase. Contrast it to thefirst order process when the outcome at time "t"-1 changes the outcome at time "t".**ee also***

Zero object

*Zero morphism

*Zero of a function

*Zero non mathematical uses.

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