Let R be a commutative ring with unity and let a, b ∈R. Show that ⟨a, b⟩, the smallest ideal of

Let R be a commutative ring with unity and let a, b ∈R. Show...

Key Concepts

Commutative Ring with Unity

A commutative ring with unity is an algebraic structure consisting of a set equipped with two binary operations: addition and multiplication. The ring is commutative under multiplication, meaning that ab = ba for all elements a and b, and it contains a multiplicative identity (unity), typically denoted as 1, such that 1 · a = a for every element a in the ring. This structure provides the foundational context for discussing ideals and other algebraic constructs.

Ideal

An ideal in a ring is a special subset that is closed under addition and is stable under multiplication by any element from the ring. This means that if you take any two elements in the ideal, their sum is also in the ideal, and if you multiply any element from the ring with any element of the ideal, the result remains inside the ideal. Ideals are crucial in algebra because they allow the construction of quotient rings and understanding the ring's structure.

Generated Ideal

A generated ideal, especially one generated by a set of elements from a ring, is defined as the smallest ideal that contains those elements. It is constructed as the intersection of all ideals that include the given set. For elements a and b in a ring, the ideal generated by them is the collection of all possible linear combinations of a and b with coefficients from the ring, making it the minimal ideal that encompasses a and b.

Linear Combination

A linear combination in the context of a ring is an expression formed by multiplying each element of a set by some element (coefficient) from the ring and then adding the results together. For instance, for elements a and b in a ring, any element of the form ra + sb, where r and s are elements of the ring, is considered a linear combination of a and b. This concept is fundamental in understanding how ideals generated by these elements are built, as every element of the ideal must be expressible in this form.

Transcript

00:01 So in part a, they want us to show for a post set, there is exactly one maximum element given there is a greatest element.

00:07 So let's go ahead and see what we can do.

00:10 So i'm going to call the post set s less than or equal to.

00:14 All right, so prove.

00:17 So first assume that s less than an equal to is a post set.

00:26 All right.

00:27 So now let's think about what we want to do next.

00:31 Well i would say most of the time when there is something saying you want to show there is exactly one of something you should do it by way of contradiction and assume there is at least two and so that will usually be the right path or at least the path of least least least resistance normally to show this so now we should tell the rear we're doing this by way of contradiction so that bwoc just means by way of contradiction so assume that is a post -set now by way of contradiction assume that there are so our contradiction that we're going to want is there is at least two elements so or what we're going to put here is there are at least two maximal elements and a greatest element.

01:52 Okay.

01:55 So let's go ahead and define what these are.

01:58 So if it's a greatest element, it's definitely maximal.

02:03 So let's just do two of these.

02:05 So let x, y, be two, or actually, so let, so i'll say first let x be our greatest, element and y a maximal element with x not equal to y.

02:47 So now let's draw a host diagram out to maybe give us some inspiration on what to do.

02:54 So if x is our greatest element, that means it would be at the very top of this.

02:59 So like let's say we have y, z, i know, a, b, c, d, something kind of like this.

03:05 And then the hoss diagram just kind of has like this thing going on.

03:09 So if x is maximal, this means everything should be dividing or be larger.

03:17 Let me try that again.

03:18 If x is our greatest element, then everything in the set should be less than or equal to this.

03:27 But over here, we said this is maximal.

03:33 So nothing should be larger than this element.

03:37 And so this is where our contradiction is going to come up, because we're assuming over here that x is not equal to y.

03:47 All right, so let's see...

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