Question

Let \(I\) be an ideal of \(R,\) and \(S\) a subring of \(R .\) Show that \(I+S\) is a subring of \(R,\) and that \(I\) is an ideal of \(I+S\).

Short answer

In this exercise, we are given an ideal \(I\) of a ring \(R\) and a subring \(S\) of \(R\). We are asked to prove that: 1) The set \(I+S\) is a subring of \(R\). 2) \(I\) is an ideal of the subring \(I+S\). We approached this problem by first recalling the definitions of an ideal, a subring, and an ideal of a subring. Then, we showed that the set \(I+S\) is a subring of \(R\) by verifying that it is a nonempty subset of \(R\), closed under addition and multiplication, and contains the additive inverse for every element in \(I+S\). Finally, we showed that \(I\) is an ideal of \(I+S\) by demonstrating that it is a nonempty subset of \(I+S\) and satisfies being closed under addition and product with elements from \(I+S\). Based on our analysis, we can conclude that \(I+S\) is a subring of \(R\) and \(I\) is an ideal of the subring \(I+S\).

Step-by-step solution

Definitions

Before we begin, let's recall the definitions of an ideal, a subring, and an ideal of a subring: - A subring \(S\) of a ring \(R\) is a nonempty subset of \(R\) that is closed under addition and multiplication, and contains the additive inverse for every element in \(S\). - An ideal \(I\) of a ring \(R\) is a nonempty subset of \(R\) that is closed under addition and contains the product of elements in \(R\) and \(I\) (i.e., \(rI \subseteq I\) for all \(r \in R\)).

Show that \(I+S\) is a subring of \(R\)

To show that \(I+S\) is a subring of \(R\), we need to show that it is a nonempty subset of \(R\), closed under addition and multiplication, and contains the additive inverse for every element in \(I+S\). 1. Nonempty subset of \(R\): Since \(I\) and \(S\) are nonempty subsets of \(R\), \(I+S\) is also nonempty and a subset of \(R\). 2. Closed under addition: Consider any two elements \(a,b \in I+S\). Then, \(a = i_1 + s_1\) and \(b = i_2 + s_2\) for some \(i_1,i_2 \in I\) and \(s_1,s_2 \in S\). We need to show that \(a + b \in I + S\). We have \(a + b = (i_1 + s_1) + (i_2 + s_2) = (i_1 + i_2) + (s_1 + s_2)\). Since \(I\) and \(S\) are both closed under addition, \(i_1 + i_2 \in I\) and \(s_1 + s_2 \in S\). Thus, \(a + b \in I + S\). 3. Closed under multiplication: Consider any two elements \(a,b \in I+S\). Then, \(a = i_1 + s_1\) and \(b = i_2 + s_2\) for some \(i_1,i_2 \in I\) and \(s_1,s_2 \in S\). We need to show that \(a \cdot b \in I + S\). We have \(a \cdot b = (i_1 + s_1) \cdot (i_2 + s_2) = (i_1i_2 + i_1s_2 + i_2s_1) + (s_1s_2)\). Since \(I\) is an ideal, \(i_1s_2, i_2s_1 \in I\) and since \(S\) is a subring, \(s_1 s_2\in S\). Also \(i_1i_2\in I\) as \(I\) is closed under multiplication. Therefore \(a \cdot b \in I + S\). 4. Contains additive inverse: Let \(a \in I + S\). Then, \(a = i + s\) for some \(i \in I\) and \(s \in S\). Since \(S\) is a subring, it contains the additive inverse \(-s\) of \(s\). We have \(-a = -(i+s) = -i + (-s)\). Since \(I\) is an ideal, it contains the additive inverse \(-i\) of \(i\). Therefore, \(-a \in I + S\). From these four conditions, we can conclude that \(I+S\) is a subring of \(R\).

Show that \(I\) is an ideal of \(I+S\)

To show that \(I\) is an ideal of \(I+S\), we need to demonstrate that it is a nonempty subset of \(I+S\) and satisfies two conditions: closed under addition and product with elements from \(I+S\). 1. Nonempty subset of \(I+S\): Since \(I\) is a nonempty subset of \(R\), it is also a nonempty subset of its subring \(I+S\). 2. Closed under addition: Let \(i_1, i_2 \in I\). Then, \(i_1 + i_2 \in I\) since \(I\) is closed under addition, and therefore also in \(I + S\). 3. Product with elements from \(I+S\): Let \(i \in I\) and \(x \in I+S\). Then, \(x = i' + s\) for some \(i' \in I\) and \(s \in S\). We need to show that \(i\cdot x \in I\). We have \(i\cdot x = i(i' + s) = ii' + is\). Since \(I\) is an ideal, \(ii' \in I\) and \(is \in I\), and since \(I\) is closed under addition, \(ii'+is \in I\). Therefore, \(i\cdot x\in I \subseteq I + S\). From these three conditions, we can conclude that \(I\) is an ideal of \(I+S\).

Key concepts

Subring Fundamentals

A subring is a fundamental concept in ring theory, which is itself a key area of abstract algebra. To understand a subring, imagine it as a smaller 'ring' within a larger one. What makes it a 'subring' is that it adheres to the same rules of operation as the larger ring. Think of a ring as a playground where certain rules of game are followed - a subring is like a smaller circle inside this playground where players agree to follow the same set of rules.

More formally, a subring is a subset of a ring that is not empty and must satisfy three main properties related to addition and multiplication: closure under addition, closure under multiplication, and the presence of additive inverses. Closure means that performing an operation (addition or multiplication) with elements inside the subring results in an answer that is also inside the subring. For example, if two students in the playground are already friends (belong to the subring), and they befriend each other's friends, the expanding group is still within the playground's friendship circle (the subring).

The additive inverse is a fancy term for 'opposite'. If you've got 5 marbles, your additive inverse is -5 marbles, because if you combine them, you end up with zero, or no marbles. In a subring, for every marble count you have, there needs to be an opposite count so that everyone can end up with no marbles, keeping the game fair and balanced.

The Core of Ring Theory

At the heart of the exercise lies ring theory, an area in mathematics where we study rings - abstract objects that generalize many familiar mathematical structures like integers, polynomials, and matrices. A 'ring' is not about round shapes but about collections of elements that behave in a certain structured way when you add or multiply them.

In a ring, you can add and multiply elements much like you do with numbers, but subtraction and division might not always work the same way. The ring must have two important properties: it must be closed under addition and multiplication, meaning if you pick any two elements from the ring and either add or multiply them together, the result will still be in the ring. For example, consider a box of colored balls. If every time you took two balls out and put them back as one combined ball of a new color, and that new color was still one available in the original box, then your box is rather like a ring.

Also, every element must have an additive inverse, so for every color ball, there's another that combines with it to make a 'neutral' color (like combining colors to get white). Rings form the foundational framework in which subrings and ideals operate, setting the stage for more complex algebraic structures studied in higher mathematics.

Understanding the Additive Inverse

An 'additive inverse' in ring theory is like the understanding that for every action, there’s an equal and opposite reaction. It’s a simple concept: for any quantity you have, the additive inverse is the quantity that, when added to your original number, results in zero - the neutral element in addition.

For instance, if you earn \(20 (that's your 'element') then spending \)20 is like finding its additive inverse because you're back to having no extra money, or zero. In a ring, every element must have an additive inverse, ensuring that the ring always balances out nicely. This concept is crucial when examining subrings, as the existence of additive inverses for every element guarantees that you can always 'undo' an addition, keeping the structure internally consistent.

The Essence of Ring Closure Properties

Ring closure properties refer to a ring or subring's ability to 'keep things inside the family'. These properties ensure that when you perform operations like addition or multiplication within the ring, you won't suddenly find yourself outside of it. It’s like making sure that when children play tag within a playground, nobody ends up in the street.

There are two main types of closure properties we look at in a ring: closure under addition and closure under multiplication. Closure under addition means that if you take any two elements in the ring and add them, the sum is also an element of the ring. Imagine you're blending two paint colors from a set; if the result is still one of the original set colors, your paint set is 'closed' under mixing.

The closure under multiplication follows a similar logic. When multiplying two elements of the ring together, the product must be an element of the ring, too, like combining two ingredients from a recipe to make a dish that's still part of the meal plan. In the context of our exercise, showing that a set has ring closure properties helps verify that it's indeed a subring; it's like proving that all the flavors in a new ice cream offering are still within the tastes expected at your favorite ice cream shop.