Commutative rings with identity

In summary, the conversation discusses the difficulty of proving that a finite commutative ring with no zero divisors must have an identity with respect to multiplication. The suggestion is to take a nonzero element in the ring and use finiteness to show that it is the multiplicative identity. Another approach is to define a map and use its injectivity and surjectivity to show that a certain element is the identity.
  • #1
Marinela
1
0
I have a trouble proving that a finate (nonzero) commutative ring with no zero divisors must have an identity with respect to multiplication. Could anybody please give me some hints?
I do know all the definitions (of ring, commutative ring, zero divisors, identity) but have no idea how to go from there.
Thanks!
 
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  • #2
Take some nonzero element a in your ring, and look at its various powers: a, a^2, a^3, ... Now use the fact that the ring has finitely many elements.

Edit:
Maybe this is not the easiest approach. It might be cleaner if you define a map x->ax on your ring. This is an injection, and hence a surjection (by finiteness). In particular, a=ax', for some x in the ring. Claim: x' is the multiplicative identity.
 
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  • #3


It is true that a finite nonzero commutative ring with no zero divisors must have an identity with respect to multiplication. To prove this, we can use the fact that a finite commutative ring is a field if and only if it has no zero divisors. Since our ring has no zero divisors, it must be a field. And in a field, every nonzero element has a multiplicative inverse. This means that for every element a in our ring, there exists an element b such that ab=1, where 1 is the identity element. This shows that our ring must have an identity with respect to multiplication.

Another way to approach this proof is to use the fact that in a finite commutative ring, every nonzero element is either a unit (has a multiplicative inverse) or a zero divisor. Since our ring has no zero divisors, every nonzero element must be a unit. This means that every element in our ring has a multiplicative inverse, and thus our ring must have an identity element with respect to multiplication.

In either approach, the key is to use the fact that our ring is finite and has no zero divisors. This allows us to conclude that every nonzero element must be a unit, and therefore, the ring must have an identity element with respect to multiplication. I hope these hints help you in your proof.
 

1. What is a commutative ring with identity?

A commutative ring with identity is a mathematical structure that consists of a set with two binary operations, addition and multiplication, that satisfy certain properties. The addition operation makes the set into an abelian group, and the multiplication operation is associative and commutative. The set also contains two special elements, 0 and 1, which act as the additive and multiplicative identities, respectively.

2. How is a commutative ring with identity different from a general ring?

A commutative ring with identity differs from a general ring in that its multiplication operation is commutative. This means that the order of multiplication does not matter, whereas in a general ring, the order of multiplication can affect the result. Additionally, a commutative ring with identity has the special element 1, which acts as the multiplicative identity, while a general ring may not necessarily have this element.

3. What are some common examples of commutative rings with identity?

Some common examples of commutative rings with identity include the set of integers (ℤ) with addition and multiplication as the binary operations, the set of real numbers (ℝ) with addition and multiplication, and the set of polynomials in one variable (ℤ[x]) with addition and multiplication. These structures all satisfy the properties of a commutative ring with identity.

4. Can a commutative ring with identity have more than one multiplicative identity?

No, a commutative ring with identity can only have one multiplicative identity. This is because the multiplicative identity must satisfy the property that a · 1 = 1 · a = a for all elements a in the ring. If there were more than one multiplicative identity, this property would not hold.

5. How are commutative rings with identity used in mathematics?

Commutative rings with identity are used in various branches of mathematics, including algebra, number theory, and geometry. They are important in studying properties of numbers, polynomials, and other mathematical structures. They also have applications in cryptography, coding theory, and other areas of computer science.

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