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The multiplicative invertible elements in the remaining class ring Z12 of module 2 are: 1, 5, 7, 11.

In the remaining class ring Z12 of module 2, any integer a if it is coprime with 12, then a exists multiplicative inverse in Z12, that is, there exists b such that ab≡1(mod 12). According to Euler's theorem, when a and 12 are prime, there is a^φ(12) ≡ 1(mod 12), where φ(12)=4 is the Euler function of 12.

Therefore, in Z12, integers that are prime to 1, 5, 7, and 11 are multiplicative invertible elements; they are 1, 5, 7, and 11, respectively.

Set 2Z The group formed by the operation of addition is not cyclic.

a

cyclic group means that there is an element a such that every element in the group can be represented as a power of a, i.e. the group is generated from A. For 2Z, its elements are of the form 2n, where n∈Z. Suppose 2Z is a cyclic group, then there is an element a such that 2n=a^n is true for all n. However, we can find that when n is odd, the left side is even and the right side is odd and not equal, so 2Z is not a cyclic group.

3, set {(0, 0)(0 a) | a ∈ R} about addition and multiplication of matrix is not the whole ring.

A whole ring is a set that satisfies both addition and multiplicative closure, associative law, commutative law, distributive law, existence of addition and multiplicative identity element, existence of multiplicative inverse element(for non-zero elements). Here we consider respectively set {(0, 0)(0 a) | a ∈ R} that addition and multiplication in the matrix is satisfy these properties.

For addition, the sum of any two matrices is still in the set and satisfies closure. Associative law of addition, commutative law of addition, existence of the identity element of addition are all true. However, there is no additive inverse in this set, because for any matrix(0 a), there is no other matrix(0 b) such that their sum is(0 0). Therefore, the addition of this set with respect to matrices is not a group and thus does not satisfy the definition of a whole ring.

For multiplication, the product of any two matrices in the set is also still in the set and satisfies closure. But multiplication is not associative. For example, for matrix(0),(0 b),(0 c), there are(a) 0((0 b),(0 c)) =(0, 0)(0 a) =(0, 0), and((a) 0(0(0 c) b) =(0, 0)(0 c) =(0, 0), they are not equal, the product of multiplication not meet the associative law. Therefore, the multiplication of this set with respect to matrices is not an integral ring.