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Post Closed as "Not suitable for this site" by Pace Nielsen, David Handelman, Chris Godsil, LSpice, domotorp
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David Roberts ♦
David Roberts ♦
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Is the multiplication beetweenbetween even numbers an associative algebra?

We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist? It has been proposed as a counterexample the set of even numbers. We believe it is wrong because it can't be a vector space. If an associative but not unital algebra exists, how can we write its associative property? We usually write it as a commutative diagram or in terms of maps composition. In these terms the identity seams unavoidable. i.e. $$ m\circ\left(m\otimes id\right) = m\circ\left(id\otimes m\right)\qquad associativity$$$$ m\circ\left(m\otimes id\right) = m\circ\left(id\otimes m\right)\qquad \text{associativity} $$ with $m : A\otimes A\rightarrow A$$m : A\otimes A\to A$.

Is the multiplication beetween even numbers an associative algebra?

We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist? It has been proposed as a counterexample the set of even numbers. We believe it is wrong because it can't be a vector space. If an associative but not unital algebra exists, how can we write its associative property? We usually write it as a commutative diagram or in terms of maps composition. In these terms the identity seams unavoidable. i.e. $$ m\circ\left(m\otimes id\right) = m\circ\left(id\otimes m\right)\qquad associativity$$ with $m : A\otimes A\rightarrow A$

Is the multiplication between even numbers an associative algebra?

We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist? It has been proposed as a counterexample the set of even numbers. We believe it is wrong because it can't be a vector space. If an associative but not unital algebra exists, how can we write its associative property? We usually write it as a commutative diagram or in terms of maps composition. In these terms the identity seams unavoidable. i.e. $$ m\circ\left(m\otimes id\right) = m\circ\left(id\otimes m\right)\qquad \text{associativity} $$ with $m : A\otimes A\to A$.

Removed the deprecated (abstract-algebra) tag - see the tag info: https://mathoverflow.net/tags/abstract-algebra/info (if there are some other suitable tags, choose them instead.)
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Martin Sleziak
Martin Sleziak
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Is the multiplication beetween even numbers an associative algebra?  

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MicAndDim
MicAndDim
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We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist? HasIt has been proposed as a counterexample the set of even numbers. We believe it is wrong because it maybe it can't be a vector space. If an associative but not unital algebra exists, how can we write its associative property? We usually write it as a commutative diagram or in terms of maps composition. In thisthese terms the identity seams unavoidable. i.e. $$ m\circ\left(m\otimes id\right) = m\circ\left(id\otimes m\right)\qquad associativity$$ with $m : A\otimes A\rightarrow A$

We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist? Has been proposed as a counterexample the set of even numbers. We believe it is wrong because it maybe it can't be a vector space. If an associative but not unital algebra exists, how can we write its associative property? We usually write it as a commutative diagram or in terms of maps composition. In this terms the identity seams unavoidable. i.e. $$ m\circ\left(m\otimes id\right) = m\circ\left(id\otimes m\right)\qquad associativity$$ with $m : A\otimes A\rightarrow A$

We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist? It has been proposed as a counterexample the set of even numbers. We believe it is wrong because it can't be a vector space. If an associative but not unital algebra exists, how can we write its associative property? We usually write it as a commutative diagram or in terms of maps composition. In these terms the identity seams unavoidable. i.e. $$ m\circ\left(m\otimes id\right) = m\circ\left(id\otimes m\right)\qquad associativity$$ with $m : A\otimes A\rightarrow A$

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Pete L. Clark
Pete L. Clark
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MicAndDim
MicAndDim
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