If A and B are ideals of a ring, show that the product of A and B, A B ={a1 b1+a2 b2+⋯+an bn |ai

step 1 In Part A, we're asked to show that the numbers 6 and 28 are perfect. So this means that they ar

If A and B are ideals of a ring, show that the product of A...

Key Concepts

Ring

A ring is an algebraic structure consisting of a set equipped with two binary operations, typically called addition and multiplication, that satisfy properties such as associativity, the existence of an additive identity, and distributivity of multiplication over addition. Rings serve as the foundational setting for studying algebraic properties and structures, including ideals.

Ideal

An ideal is a special subset of a ring that is not only an additive subgroup but also absorbs multiplication by any element of the ring. This means if I is an ideal and r is any ring element while i is any element of I, then both ri and ir are in I. Ideals are central in constructing quotient rings and analyzing ring properties.

Product of Ideals

The product of two ideals in a ring is defined as the set of all finite sums of products of elements, where each product consists of one element from each ideal. This concept uses the distributive property of the ring and is instrumental in understanding how different ideals interact within the ring structure.

Closure Properties

To verify that a set qualifies as an ideal, it must be closed under addition and under multiplication by arbitrary ring elements. In the context of the product of ideals, demonstrating that the set remains closed under these operations—specifically, that multiplying any element of the ring with an element of the product yields another element from the product—is essential for establishing its status as an ideal.

Transcript

00:03 In part a, we're asked to show that the numbers 6 and 28 are perfect.

00:11 So this means that they are equal to the sum of the positive dividers other than themselves.

00:18 So consider 6.

00:25 The divisors of 6 include 1, 2, 3, and 6, and the divisors of 28 include 1, 2, 3, and 6, to 4, 7, 14, and 28.

01:01 Now let's find the sum of all positive divisors other than the integer itself.

01:06 We have that 1 plus 2 plus 3 is indeed 6, and that 1 plus 2 plus 4 plus 7 plus 14.

01:18 This is going to be 14 plus 7 plus 4 is 25, plus 2 is 27, plus 1 is 20.

01:24 8, which is what we wanted.

01:26 So this shows that 6 and 28 are perfect numbers.

01:38 Okay, so just an example there.

01:40 Now in part b, we're as to generalize and show that 2 to the p minus 1 times 2 to the p minus 1 is a perfect number whenever 2 to the p minus 1 is prime.

01:58 So we're going to suppose that 2 to the p minus 1 is prime.

02:15 Let's determine all the divisors of the number.

02:18 2 to the p minus 1 times 2 to the p minus 1.

02:26 So the divisors of 2 to the p minus 1 times 2 to the p minus 1.

02:41 Well these include, of course, 1 as well as 2, since 2 divides 2 to the p minus 1 and 2 squared, all the way up to 2 to the p minus 1.

03:03 All these numbers divide 2 to the p minus 1, which is a factor of our number.

03:08 And what else can we include? well, we could also include the factor 2 to the p minus 1, 2 to the p minus 1, since this is a prime factor...