Absorbing element: Difference between revisions
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In [[algebra]], an '''absorbing element''' or a '''zero element''' for a [[binary operation]] has a property similar to that of [[multiplication]] by [[zero]]. | In [[algebra]], an '''absorbing element''' or a '''zero element''' for a [[binary operation]] has a property similar to that of [[multiplication]] by [[zero]]. | ||
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* The zero (additive identity element) of a [[ring (mathematics)|ring]] is an absorbing element for the ring multiplication. | * The zero (additive identity element) of a [[ring (mathematics)|ring]] is an absorbing element for the ring multiplication. | ||
* The [[zero matrix]] is the absorbing element for [[matrix multiplication]]. | * The [[zero matrix]] is the absorbing element for [[matrix multiplication]]. | ||
* The [[empty set]] is the absorbing element for [[intersection]] of sets. | * The [[empty set]] is the absorbing element for [[intersection]] of sets.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:00, 5 July 2024
In algebra, an absorbing element or a zero element for a binary operation has a property similar to that of multiplication by zero.
Formally, let be a binary operation on a set X. An element O of X is absorbing for if
holds for all x in X. An absorbing element, if it exists, is unique.
Examples
- The zero (additive identity element) of a ring is an absorbing element for the ring multiplication.
- The zero matrix is the absorbing element for matrix multiplication.
- The empty set is the absorbing element for intersection of sets.