Commutative Rings and Ideals

This question asks about properties of commutative rings with unity, specifically focusing on the existence and number of maximal and prime ideals. A ring $R$ is commutative if $ab = ba$ for all $a, b \in R$. A ring has unity if there exists an element $1 \in R$ such that $1 \cdot a = a \cdot 1 = a$ for all $a \in R$. Ideals are special subsets of a ring that behave well with ring operations.

Maximal and Prime Ideals

Let's define the types of ideals mentioned:

• A proper ideal $M$ of a commutative ring $R$ with unity is called a maximal ideal if the only ideals containing $M$ are $M$ itself and $R$.
• A proper ideal $P$ of a commutative ring $R$ is called a prime ideal if for any $a, b \in R$, if $ab \in P$, then $a \in P$ or $b \in P$.

In a commutative ring with unity, there is a crucial relationship between maximal and prime ideals:

• Every maximal ideal is a prime ideal.

Also, Krull's Theorem states:

• Every proper ideal in a commutative ring with unity is contained in a maximal ideal. A direct consequence is that a commutative ring with unity must have at least one maximal ideal.

Analyzing the Options

Let's examine each statement in the options based on these facts about commutative rings with unity R:

Option 1: R may have no maximal ideals.

According to Krull's Theorem, every commutative ring with unity has at least one maximal ideal. Therefore, this statement is false for any commutative ring R with unity.

Option 3: R can have one or more maximal ideals but no prime ideals.

We know that every maximal ideal in a commutative ring with unity is also a prime ideal. Since a commutative ring with unity must have at least one maximal ideal, it must also have at least one prime ideal. Therefore, this statement is false for any commutative ring R with unity.

Option 4: R has at least two prime ideals.

Consider a field, like the rational numbers $\mathbb{Q}$ or the real numbers $\mathbb{R}$. A field is a commutative ring with unity. The only ideals in a field are the zero ideal $\{0\}$ and the field itself. The zero ideal $\{0\}$ is a maximal ideal (because $R/\{0\} \cong R$, which is a field, and $R$ is not the zero ring) and thus also a prime ideal. $R$ is not a proper ideal. So, a field has exactly one maximal ideal (the zero ideal) and exactly one prime ideal (the zero ideal). Therefore, it is not necessarily true that R has at least two prime ideals. This statement is false for fields.

Option 2: R can have exactly two maximal ideals.

While the question asks what is "necessarily true", given that options 1, 3, and 4 are demonstrably false for *any* or *some* commutative rings with unity, this option states something that *can* be true. Consider the ring of integers modulo 6, $\mathbb{Z}_6$. This is a commutative ring with unity. The ideals of $\mathbb{Z}_6$ are $(0)$, $(2)=\{0, 2, 4\}$, $(3)=\{0, 3\}$, and $(1)=\{0, 1, 2, 3, 4, 5\}=\mathbb{Z}_6$. The proper ideals are $(0)$, $(2)$, and $(3)$.

• The ideal $(2)$ is maximal because $\mathbb{Z}_6 / (2) \cong \mathbb{Z}_2$, which is a field.
• The ideal $(3)$ is maximal because $\mathbb{Z}_6 / (3) \cong \mathbb{Z}_3$, which is a field.
• The ideal $(0)$ is not maximal because it is properly contained in $(2)$ and $(3)$.

Thus, $\mathbb{Z}_6$ has exactly two maximal ideals: $(2)$ and $(3)$. This shows that it is possible for a commutative ring with unity to have exactly two maximal ideals. Since options 1, 3, and 4 are not necessarily true (indeed, 1 and 3 are necessarily false, and 4 is false for fields), the statement that R can have exactly two maximal ideals is the only one that represents a possible scenario for a commutative ring with unity among the choices, and is thus the most likely intended correct answer in this context.

Conclusion

Based on the analysis, options 1 and 3 are necessarily false for any commutative ring with unity. Option 4 is false for fields (which are commutative rings with unity). Option 2 describes a property that is possible for a commutative ring with unity (e.g., $\mathbb{Z}_6$). Therefore, among the given statements, "R can have exactly two maximal ideals" is the only one that can be true for a commutative ring with unity.