Question

Let \(R\) be a ring, and let \(\left\\{I_{i}\right\\}_{i=0}^{\infty}\) be a sequence of ideals of \(R\) such that \(I_{i} \subseteq I_{i+1}\) for all \(i=0,1,2, \ldots .\) Show that the union \(\bigcup_{i=0}^{\infty} I_{i}\) is also an ideal of \(R .\)

Short answer

Question: Show that the union of an increasing sequence of ideals is also an ideal. Answer: To show that the union of an increasing sequence of ideals is also an ideal, we need to check two main properties: closure under subtraction and closure under left and right multiplication. By considering any two elements in the union and showing that their subtraction is also in the union, we establish closure under subtraction. Similarly, by considering any element in the union and any element in the ring, we show that their product (left and right multiplication) is also in the union, thus establishing closure under multiplication. Consequently, the union is indeed an ideal.

Step-by-step solution

Closure under subtraction

Consider any two elements \(a, b\) in the union \(\bigcup_{i=0}^{\infty} I_{i}\). By definition of union, there exist indices \(m, n\) such that \(a \in I_{m}\) and \(b \in I_{n}\). Without loss of generality, assume that \(m \leq n\). Then, we have \(I_{m} \subseteq I_{n}\). Therefore, both \(a\) and \(b\) are in \(I_{n}\). As \(I_{n}\) is an ideal, it is closed under subtraction. Consequently, we have \(a - b \in I_{n}\). Since \(a - b \in I_{n}\), it is also in the union, i.e., \(a - b \in \bigcup_{i=0}^{\infty} I_{i}\). Thus, the union is closed under subtraction.

Closure under left and right multiplication

Now let \(r \in R\) and \(a \in \bigcup_{i=0}^{\infty} I_{i}\). Then, there exists some index \(m\) such that \(a \in I_{m}\). Since \(I_{m}\) is an ideal, it is closed under left and right multiplication by elements of \(R\). Therefore, we have \(ra \in I_{m}\) and \(ar \in I_{m}\). Since \(ra, ar \in I_{m}\), they are also in the union, i.e., \(ra, ar \in \bigcup_{i=0}^{\infty} I_{i}\). Thus, the union is closed under left and right multiplication by elements of \(R\).

Conclusion

We have shown that the union \(\bigcup_{i=0}^{\infty} I_{i}\) is closed under subtraction and left and right multiplication by elements of \(R\). Therefore, the union is indeed an ideal of \(R\).

Key concepts

Closure Under Subtraction

Understanding the property of closure under subtraction is fundamental in ring theory. It is a condition required for a subset of a ring to qualify as an ideal. If you take any two elements from this subset and subtract one from the other, the result should also be a member of that subset. In plain terms, it stays 'inside' the set when performing subtraction.

For example, consider the ideal \(I_n\) from our exercise. If you pick any two elements \(a\) and \(b\) from \(I_n\), their difference \(a - b\) must also be in \(I_n\). This concept is crucial because it ensures the structural integrity of the ideal within the ring. Without closure under subtraction, our set would not meet the definition of an ideal.

Closure Under Multiplication

Similar to closure under subtraction, closure under multiplication is another essential property for a set to be an ideal in ring theory. It demands that when an element from the ring is multiplied with any element from the ideal — whether from the left or the right — the result remains within the ideal.

To visualize this, if \(r\) is any element of the ring \(R\) and \(a\) belongs to the ideal \(I_m\), then both products \(ra\) and \(ar\) are required to be in \(I_m\). This property confirms that the ideals can absorb multiplication by the encompassing ring, which is a cornerstone for the workings and applications of ring theory.

Ideals of a Ring

In ring theory, ideals are special subsets that retain certain operations of the ring. An ideal can be thought of as the 'perfect' piece of a ring that adheres to its arithmetic rules. Specifically, for a subset of a ring to be called an ideal, it must satisfy two conditions: it must be closed under subtraction, as we discussed earlier, and it must also be closed under multiplication by any element of the ring.

These properties make ideals a central concept in the study of rings because they preserve the ring's structure and help in constructing new rings, such as quotient rings. They're like the building blocks that maintain the ring's essence during various constructions and transformations.

Ring Theory

Ring theory is a branch of abstract algebra that deals with rings, which are algebraic structures equipped with two binary operations that we often think of as addition and multiplication. These operations must satisfy certain criteria such as associativity and distributivity.

Rings can come in many flavors, such as commutative rings where the order in multiplication doesn't matter, or rings with identity where there's a special element that acts as a `1` does with numbers. Studying rings gives us insight into various mathematical systems, including integers, polynomials, and matrices, to name a few.

Ascending Chain of Ideals

The concept of an ascending chain of ideals pertains to a sequence of ideals within a ring, where each ideal is a subset of the next. This is the setup in our exercise. In mathematical terms, if \(I_0, I_1, I_2, ...\) are ideals in a ring, and \(I_0 \subseteq I_1 \subseteq I_2 \subseteq ...\), we have an ascending chain.

The significance of an ascending chain comes into play when considering the union of all these ideals. As we saw in the steps of our exercise, this union itself forms an ideal under certain conditions. This behavior of the ascending chain highlights how the infinite 'growth' of ideals can result in a larger ideal, showcasing the fascinating and sometimes counterintuitive nature of infinite constructions in mathematics.