Prove that each of the following is a subring of the indicated ring. {x 2^y: x, y ∈ℤ} is a subri

step 1 To prove that a set with the operation is a group, we need to verify these three axioms. First,

Prove that each of the following is a subring of the...

Key Concepts

Additive Inverses

A fundamental requirement for any ring is that every element must have an additive inverse within the structure. For the subset under consideration, it must be shown that for every element of the form x · 2^y, its additive inverse, which is -x · 2^y, is also in the set. This completes the necessary verification for the subset to inherit a complete ring structure, thereby forming a subring.

Closure under Addition

Checking closure under addition means demonstrating that the sum of any two elements in the subset is also an element of the subset. For a set defined by numbers of the form x · 2^y where x and y are integers, one must show that adding any two such numbers results in another number that can be written in the same form. This is a key step in verifying that the subset forms a subring.

Closure under Multiplication

Ensuring closure under multiplication involves showing that the product of any two elements in the subset remains within the subset. For elements given by x · 2^y, the multiplication of two such elements yields (x · a) · 2^(y+b), where x, y, a, and b are integers, preserving the required form. This property is essential for establishing the subring structure.

Ring

A ring is an algebraic structure consisting of a set equipped with two binary operations (commonly addition and multiplication) that satisfy properties such as associativity for both operations, distributivity of multiplication over addition, the existence of an additive identity, and the existence of additive inverses. In the context of real numbers, these properties are readily satisfied, making ? a ring, which serves as the ambient structure for studying subrings.

Subring

A subring is a subset of a ring that is itself a ring under the same operations. This means that the subset must contain the additive identity, be closed under addition and multiplication, and contain additive inverses of its elements. Proving that a set is a subring involves verifying these conditions, ensuring that the subset inherits the ring structure from the larger ring.

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