Is Z[sqrt2] defined as {a+b(sqrt2)|a,b in Z}?

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Z[sqrt(2)] is defined as the smallest ring containing all of Z and sqrt(2), and can be expressed as {a + b(sqrt(2)) | a, b in Z}. This set satisfies the properties of a ring, as it is closed under both addition and multiplication, given that the commutative, associative, and distributive laws hold for all real numbers. In contrast, Z[cuberoot(2)] cannot be expressed in the same form {a + b(cuberoot(2)) | a, b in Z} because it fails to meet the closure property under multiplication.

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Is Z[sqrt2] defined as {a+b(sqrt2)|a,b in Z}?
 
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No, it's not defined that way. Since Z is a ring (but not a field), Z[sqrt(2)] is defined as "the smallest ring containing all of Z and sqrt(2)".

Your question really is- can that be expressed as {a+ b(sqrt(2))| a,b in Z}.

To answer that question, you have to see if it is a ring. Certainly, since rings are closed under multiplication, it must contain b(sqrt(2)) for any integer b. Since rings are closed under addition, it must also contain any number of the form a+ b(sqrt(2)). Is that enough? Is this closed under addition? Is it closed under multiplication? Since the commutative, associative and distributive laws are true for all real numbers and this set is clearly a subset of the real numbers, you don't need to show those separately.
 
Yes {a+ b(sqrt(2))| a,b in Z} does satisfy being a ring so Z[sqrt2] can be expressed as that set.
 
Good. What about Z[cuberoot(2)]? Is that equal to {a+ bcuberoot(2) |a, b in Z}?
 
No because it is not closed under multiplication.
 

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