Pretty much. Fields are generally considered to be the structure that links the common notion addition and multiplication. In particular, you need 3 things to qualify as a field:
1) Multiplication behaves as expected (without reference to addition)
2) Addition behaves as expected (without reference to multiplication)
3) a(b+c) = ab + bc
The third requirement is the only part of fields that links multiplication to addition. In particular, if we assume that 2=1+1, then we have: 2a = (1+1)a = a(1+1) = a+a
1) Multiplication behaves as expected (without reference to addition)
2) Addition behaves as expected (without reference to multiplication)
3) a(b+c) = ab + bc
The third requirement is the only part of fields that links multiplication to addition. In particular, if we assume that 2=1+1, then we have: 2a = (1+1)a = a(1+1) = a+a