Noetherian Ring

CONDITION 1:Let A is contained in B be aunitary commutative ring extension such that U(B) represents the set of units of B.For each b belong to B there exist d belong to U(B) and a belong to A such that b=da. NOTE:The conductor of A in B is the largest common ideal A:B={a belongs to A:aB is subset of A} of A and B.PROBLEM:Let A is contained in B be a domain extension which satisfies the condition 1 and M=A:B be a maximal ideal in A.Then B is Noetherian iff A is Noetherian.Moreover give an example

1)What does it mean "domain extension"? Does it simply mean ? And so B is an extension of A?
2)Is A a commutative ring too? Or just a subset?