Zero element: Difference between revisions

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In [[mathematics]], '''zero element''' may refer to:
{{subpages}}


* The [[neutral element]] of a [[group (mathematics)|group]] or [[monoid]] written in additive notation.
In algebra, the term '''zero element''' is used with two meanings,
* An [[absorbing element]] of a binary operation.
both in analogy to the number [[zero (mathematics)|zero]].


{{disambig}}
* For an additively written binary operation, ''z'' is a ''zero element'' if (for all ''g'')
:: ''z'' + ''g'' = ''g'' = ''g'' + ''z''
: i.e., it is the (unique) [[neutral element]] for this operation.
 
* For a multiplicatively written binary operation, ''a'' is a ''zero element'' if (for all ''g'')
::  ''ag'' = ''g'' = ''ga''
: i.e., it is the (unique) [[absorbing element]] for this operation.
 
In rings (not only the real or complex numbers) 0 is the zero element in both senses.
 
In addition to these "two-sided" zero elements, (one-sided) ''left'' or ''right'' zero elements are also considered
for which only one of the two identities is valid for all ''g''.
 
One-sided zero elements need not be unique.

Latest revision as of 19:23, 10 November 2009

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In algebra, the term zero element is used with two meanings, both in analogy to the number zero.

  • For an additively written binary operation, z is a zero element if (for all g)
z + g = g = g + z
i.e., it is the (unique) neutral element for this operation.
  • For a multiplicatively written binary operation, a is a zero element if (for all g)
ag = g = ga
i.e., it is the (unique) absorbing element for this operation.

In rings (not only the real or complex numbers) 0 is the zero element in both senses.

In addition to these "two-sided" zero elements, (one-sided) left or right zero elements are also considered for which only one of the two identities is valid for all g.

One-sided zero elements need not be unique.