I am studying Linear algebra from a book called "The Linear Algebra a Beginning Graduate Student Ought to Know " by Jonathan S. Golan.
And in the definition of a field the book mentioned that the multiplication should be distributed over addition from left only. But in Hungerford "Abstract Algebra" the author said that multiplication should be distributed over addition from left and right. I wondering why this condition differs from one book to the other? does distribution of multiplication over addition from left implies its distribution over addition from right? could anyone explains this point for me please?
In any ring both distributive laws hold, whether or not multiplication is commutative. (https://en.wikipedia.org/wiki/Ring_(mathematics)#Definition)
In a field, multiplication is commutative, so it suffices to require just one of the distributive laws.
In Golan's linear algebra the set of scalars is a field. In Hungerford's more general text on abstract algebra I suspect he is axiomatizing a ring.