if there is one output for every input

Yes, but all relations are not functions.

Sure. There are other applications of arithmetic, but algebra without arithmetic is impossible. A broad knowledge of arithmetic is essential for mastery of algebra.

P. M. Cohn has written:

'Skew field construction' -- subject(s): Rings (Algebra)

'Algebra' -- subject(s): Abstract Algebra, Algebra, Algebra, Abstract

'Free ringsand their relations' -- subject(s): Associative rings

'Skew field constructions' -- subject(s): Division rings, Algebraic fields

'Algebra. Volume 2.'

'Algebraic numbers and algebraic functions' -- subject(s): Algebraic functions, Algebraic fields

Yes, but all relations are not functions.