Zero element: Difference between revisions
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In algebra, the term '''zero element''' is used with two meanings, | |||
both in analogy to the number [[zero (mathematics)|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. | |||
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Latest revision as of 12:00, 10 November 2024
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.