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I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...
I need help with some remarks of Bresar on the centre of an algebra ...
Commencing a section on Central Algebras, Bresar writes the following:https://www.physicsforums.com/attachments/6243
In the above text we read the following:
" ... The center of a unital algebra obviously contains scalar multiples of unity ... ... "Now the center of a unital algebra $$A$$ is defined as the set $$Z(A)$$ such that
$$Z(A) = \{ c \in A \ | \ cx = xc \text{ for all x } \in A \} $$Now ... clearly $$1 \in Z(A)$$ since $$1x = x1$$ for all $$x$$ ...
BUT ... why do elements like $$3$$ belong to $$Z(A)$$ ... ?
That is ... how would we demonstrate that $$3x = x3$$ for all $$x \in A$$ ... ?
Hope someone can help ...
Peter
I need help with some remarks of Bresar on the centre of an algebra ...
Commencing a section on Central Algebras, Bresar writes the following:https://www.physicsforums.com/attachments/6243
In the above text we read the following:
" ... The center of a unital algebra obviously contains scalar multiples of unity ... ... "Now the center of a unital algebra $$A$$ is defined as the set $$Z(A)$$ such that
$$Z(A) = \{ c \in A \ | \ cx = xc \text{ for all x } \in A \} $$Now ... clearly $$1 \in Z(A)$$ since $$1x = x1$$ for all $$x$$ ...
BUT ... why do elements like $$3$$ belong to $$Z(A)$$ ... ?
That is ... how would we demonstrate that $$3x = x3$$ for all $$x \in A$$ ... ?
Hope someone can help ...
Peter