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# 数学代写|交换代数代写Commutative Algebra代考|MATH765 Principal Ideal Domains, Euclidean Rings

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## 数学代写|交换代数代写Commutative Algebra代考|Principal Ideal Domains, Euclidean Rings

– Let A be a commutative ring.
Recall that an ideal of $\mathrm{A}$ is said to be principal if it is of the form $a \mathrm{~A}$, for some $a \in \mathrm{A}$. One also writes $(a)$ for $a \mathrm{~A}$.

Let $a, b \in \mathrm{A}$. The ideal $a \mathrm{~A}$ is contained in the ideal $b \mathrm{~A}$ if and only if there exists an element $c \in \mathrm{A}$ such that $a=b c$, that is, if and only if $b$ divides $a$.
Assume moreover that $\mathrm{A}$ is an integral domain. Let $a, b \in \mathrm{A}$ be such that $a \mathrm{~A}=b \mathrm{~A}$. Then, there exist $c, d \in \mathrm{A}$ such that $a=b c$ and $b=a d$, hence $a=a(c d)$ and $b=(c d)$. If $a \neq 0$ then $b \neq 0$; simplifying by $a$, we get $c d=1$, hence $c$ and $d$ are invertible. In other words, two non-zero elements $a$ and $b$ of an integral domain A generate the same ideal if and only if there exists a unit $u \in \mathrm{A}$ such that $b=a u$.

The units of the ring $\mathbf{A}=\mathbf{Z}$ are $\pm 1$; it is thus customary to choose, as a generator of a principal ideal, a positive element. Similarly, the units of the ring $K[X]$ of polynomials in one indeterminate $X$ and with coefficients in a field $\mathrm{K}$ are the non-zero constant polynomials and we then often choose $a$ monic polynomial for a generator of a non-zero ideal (see example $1.4 .4$ ).

## 数学代写|交换代数代写Commutative Algebra代考|Greatest common divisor, least common multiple

Greatest common divisor, least common multiple – Let A be a principal ideal domain. Let $\left(a_i\right)$ be a family of elements of $\mathrm{A}$. By the assumption on $\mathrm{A}$, the ideal I generated by the $\left(a_i\right)$ is generated by one element, say $a$. It follows that $d$ divides $a_i$ for any $i: d$ is a common divisor of all of the $a_i$. Moreover, if $d^{\prime}$ is a common divisor of the $a_i$, then $a_i \in\left(d^{\prime}\right)$ for every $i$, hence $\mathrm{I} \subset\left(d^{\prime}\right)$ and $d^{\prime}$ divides $d$. One says that $d$ is a greatest common divisor $(\mathrm{gcd})$ of the $a_i$. The word “greatest” has to be understood in the sense of divisibility: the common divisors of the $a_i$ are exactly the divisors of their $g c d$. There is in general no preferred choice of a greatest common divisor, all differ by multiplication by a unit in A.

Let $J$ be the intersection of the ideals $\left(a_i\right)$ and let $m$ be a generator of the ideal J. For any $i, m \in\left(a_i\right)$, that is, $m$ is a multiple of $a_i$ for every $i$. Moreover, if $m^{\prime} \in \mathrm{A}$ is a multiple of $a_i$ for every $i$, then $m^{\prime} \in\left(a_i\right)$ for every $i$, hence $m^{\prime} \in(m)$ and $m^{\prime}$ is a multiple of $m$. One says that $m$ is a least common multiple $(\mathrm{lcm})$ of the $a_i$. Again, the word “least” has to be understood in the sense of divisibility. As for the gcd, there is no preferred choice and all least common multiples differ by multiplication by a unit in $\mathrm{A}$.

As explained above, when $\mathrm{A}=\mathrm{Z}$ is the ring of integers, one may choose for the gcd and the lcm the unique positive generator of the ideal generated by the $a_i$, resp. of the intersection of the $\left(a_i\right)$. Then, except for degenerate cases, $d$ is the greatest common divisor and $m$ is the least common (non-zero) multiple in the naive sense too.

Similarly, when $\mathrm{A}=\mathrm{K}[\mathrm{X}]$ is the ring of polynomials in one indeterminate $X$, it is customary to choose the gcd and the $\mathrm{cm}$ to be monic polynomials (or the zero polynomial).

## 数学代写|交换代数代写COMMUTATIVE ALGEBRA代 考|PRINCIPAL IDEAL DOMAINS, EUCLIDEAN RINGS

-令 $A$ 为交换环。

，我们得到 $c d=1$ ，因此 $c$ 和 $d$ 是可逆的。换句话说，两个非零元綪 $a$ 和 $b$ 当且仅当存在一个单元时，积分域 $\mathrm{A}$ 的生成相同的理想 $u \in \mathrm{A}$ 这样 $b=a u$.

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