Algebraic varieties and irreducibility

Question

When I was reading some references I found out a conflict in the definition of an algebraic variety, one defines this notion to be irreducible and other does not Can I have a clear definition and why suppose that the variety is not neces irreducible? Thank you

Answer 1

I have always considered varieties to be irreducible over algebraically closed fields since we have tools like the Nullstellensatz at our disposal, and thus objects like coordinate rings are much nicer to evaluate in this case. In arbitrary fields, we may not have a Nullstellensatz, so there might not be much to gain by considering irreducible algebraic sets.