Some facts, better explained, from Atiyah-Macdonald

Today we shall discuss some interesting properties of elements of a ring.

1. If $a\in R$ is not a unit, then it is present in some maximal ideal of the ring $R$ . Self-explanatory.

2. If $a$ is present in every maximal ideal, then $1+xa$ is a unit for all $x\in R$ . Proof: Let $1+xa$ not be a unit. Then it is present in some maximal ideal (from 1). Let $1+xa=m$ , where $m$ is an element from the maximal ideal $1+xa$ is a part of. Then $1=m-xa$ . Hence, $1$ is also a member of the maximal ideal, which is absurd.

Let’s break down this theorem into elementary steps, and present a better proof (than given on pg.6 of “Commutative Algebra” by Atiyah-Macdonald). If $x\in M_1$ for some maximal ideal $M_1$ , then $1\pm xy\notin M_1$ for all $y\in R$ . Similarly, If $x\in M_2$ for some maximal ideal $M_2$ , then $1\pm xy\notin M_2$ for all $y\in R$ . This argument can then be extended to the fact that if $x\in$ all maximal ideals, then $1\pm xy\notin$ any maximal ideal for all $y\in R$ . An element not there in any maximal ideal is a unit. Hence, $1\pm xy$ is a unit.

3. If $1-xy$ is a unit for all $y\in R$ , then $x$ is part of every maximal ideal in $R$ . Proof: Let is assume $x$ is not part of some maximal ideal. Then there exists some $m\in$ that maximal ideal such that $m+xy=1$ . This implies that $m=1-xy$ , which is impossible as $1-xy$ is a unit. The same argument can be used for $1+xy$ .

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On pg.6 of Atiyah-Macdonald, it is mentioned that if $a,b,c$ are ideals, then $a\cap (b+c)=a\cap b+a\cap c$ if $b\subseteq a$ and $c\subseteq a$ . It is not elaborated in the book, and the flippant reader may be confused. I’d like to elaborate on this concept.

$a\cap (b+c)$ consists of those elements in $b+c$ which are also there in $a$ . Now elements of $a\cap (b+c)=[(b+c)$ wrt those elements of both $b$ and $c$ that are in $a$ ] $\bigcup$ [ $(b+c)$ wrt those elements of $b$ that are in $a$ and those elements of $c$ that are not] $\bigcup$ [ $(b+c)$ wrt those elements of $b$ that are not in $a$ and those elements of $c$ that are] $\bigcup$ [ $(b+c)$ wrt those elements of both $b$ and $c$ that are not in $a$ ]

The second and the third terms are null sets, as can be easily seen.

The fourth term is NOT necessarily empty. However, it becomes an empty set if $b,c\subseteq a$ . It may also become an empty set under other conditions which ensure that if both $b$ and $c$ are not in $a$ , then $b+c\notin a$ .

In summation, $a\cap (b+c)=a\cap b+a\cap c$ is definitely true when $b,c\subseteq a$ . However, it is also true under other conditions which ensure that the fourth term is an empty set.

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I have extended a small paragraph in Atiyah-Macdonald to a full-fledged exposition: ( $a,b,c$ and $d$ are ideals in commutative ring $R$ )

1. $a\cup (b+c)\subseteq(a\cup b)+(a\cup c)$ – Both sides contain all elements of $a$ and $b+c$ . Remember that $b\cup c\subseteq b+c$ . However, the right hand side also contains elements of the form $a+b$ and $a+c$ , which the left hand side does not contain.

2. $a\cap (b+c)$ – This has already been explained above.

3. $a+(b\cup c)=(a+b)\cup (a+c)$ – Both are exactly the same.

4. $a+ (b\cap c)\subseteq (a+b)\cap (a+c)$ – There might be $b_1,c_1\notin b\cap c$ such that $a'+b_1=a''+c_1$ . However, any element in $a+(b\cap c)$ will definitely be present in $(a+ b)\cap (a+c)$ .

5. $a(b\cup c)\supseteq ab\cup ac$ – LHS contains elements of the form $a'b_1+a''c_1$ , which the RHS doesn’t. In fact, LHS is an ideal while the RHS isn’t. You might wonder how LHS is an ideal. I have just extended the algorithm used to make $AB$ an ideal when $A$ and $B$ are both ideals to situations in which $A$ is an ideal and $B$ is any subset of $R$ .