Ring Theory Calculator

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Example

Created on 2024-06-20Asked by Chloe Hall (Solvelet student)

Determine if the set of integers

\mathbb{Z}

with standard addition and multiplication forms a ring.

Solution

To determine if the set of integers

\mathbb{Z}

with standard addition and multiplication forms a ring, we need to check the ring axioms: 1. **Closure under addition and multiplication:** - For any

a, b \in \mathbb{Z}

, both

a + b \in \mathbb{Z}

and

ab \in \mathbb{Z}

. 2. **Associativity of addition and multiplication:** - Addition:

(a + b) + c = a + (b + c) \quad \text{for all} \; a, b, c \in \mathbb{Z}.

- Multiplication:

(ab)c = a(bc) \quad \text{for all} \; a, b, c \in \mathbb{Z}.

3. **Existence of additive identity:** - There exists an element

0 \in \mathbb{Z}

such that

a + 0 = a

for all

a \in \mathbb{Z}

. 4. **Existence of additive inverses:** - For each

a \in \mathbb{Z}

, there exists an element

-a \in \mathbb{Z}

such that

a + (-a) = 0

. 5. **Distributivity of multiplication over addition:**

a(b + c) = ab + ac \quad \text{and} \quad (a + b)c = ac + bc \quad \text{for all} \; a, b, c \in \mathbb{Z}.

Since

\mathbb{Z}

satisfies all the ring axioms, it forms a ring. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Ava Lopez on Solvelet

1. Determine whether the set of even integers forms a ring under addition and multiplication.2. Find the quotient ring of the ring of integers modulo 4.,

DefinitionIn abstract algebra, ring theory is the study of rings-algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Rings are a generalization of the integers (with the addition of a+b, the multiplication of a*b, and where a*1 is a for every integer a), and include structures like fields and polynomial rings. Example: the ring of all 2×2 matrices with real number entries under matrix addition and multiplication.

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