Proving a set is a field or a ring

Hi there. I know what properties are necessary for a ring and for a field. I'm very poor at formal proofs however. Could someone help me learn how to formally prove whether a set is a ring or a field? Here are some questions, pick whichever one you would like to answer


1. Show that the following sets are rings. Determine whether they are commutative and if there is a unity.

1. Z[x], the set of all polynomials in x with integer coefficients.
2. The set of all 3x3 matrices with real entries.
3. The set of all even integers n ∈ Z.

2. Show that the set Q(√2) = { a + b√2 | a, b are rationals } forms a field.

First I'll try my best to answer one of them:
1C)
let n = 2m or 2k where m and k are integers
2m + 2k = 2(m+k) since 2(m+k) is even we have additive closure
2m * 2k = 4mk = 2(2mk) since 2(2mk) is even we have multiplicative closure
*I'm not exactly sure how to prove the other properties, or even if I have to? Can't I just assume this set has them because integers have them, and this is a set of integers? It seems obvious after determining closure that this set is both a ring and a field

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Originally Posted by archon

Hi there. I know what properties are necessary for a ring and for a field. I'm very poor at formal proofs however. Could someone help me learn how to formally prove whether a set is a ring or a field? Here are some questions, pick whichever one you would like to answer


1. Show that the following sets are rings. Determine whether they are commutative and if there is a unity.

1. Z[x], the set of all polynomials in x with integer coefficients.
2. The set of all 3x3 matrices with real entries.
3. The set of all even integers n ∈ Z.

2. Show that the set Q(√2) = { a + b√2 | a, b are rationals } forms a field.

First I'll try my best to answer one of them:
1C)
let n = 2m or 2k where m and k are integers
2m + 2k = 2(m+k) since 2(m+k) is even we have additive closure
2m * 2k = 4mk = 2(2mk) since 2(2mk) is even we have multiplicative closure
*I'm not exactly sure how to prove the other properties, or even if I have to? Can't I just assume this set has them because integers have them, and this is a set of integers? It seems obvious after determining closure that this set is both a ring and a field

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First of all, you need to know and fully understand what are the two operations defined on a particular

set that aspires to be a ring...and then you check all the axioms.

Certainy for the 3 example, say, there are several axioms that are automatic from a bigger set yours is contained in, say

commutativity (in this case of both addition and multiplication), associativity (since these two are true in ), etc., but

ANY non-obvious axiom must be fully checked.

Tonio