Archive for ‘Abstract Algebra’


An example of how to select ideals to obtain quotient rings with desirable properties

The fundamental homomorphism theorem for rings employs the kernel K of a homomorphism f:A\rightarrow B to construct a quotient ring A/K isomorphic to ring B. Conversely, the homomorphic image A/J of a ring A is useful for factoring out unwanted features of A to obtain a quotient ring A/J with the desirable properties. Selecting an appropriate ideal J is the key for acquiring the desired quotient ring A/J.

In some cases, picking a suitable ideal J can be done methodically by tracing the desired property of A/J back to an analogous property of J. To demonstrate the process of selecting J, the simple example of obtaining a quotient ring A/J that doesn’t contain any divisors of zero will be considered.

The example under consideration is stated as follows. Let A be a commutative ring with unity. An ideal J of A is prime if and only if A/J is an integral domain.

Briefly recall the associated definitions. An integral domain is a commutative ring with unity that doesn’t have any divisors of zero. An ideal  J of a commutative ring A is prime if \forall a,b\in A with ab\in J it follows that a\in J or b \in J.

It is straightforward to prove the equivalence of this example. To prove that the integral domain has no divisors, it suffices to observe that a\in J \Leftrightarrow J=J+a. Emphasis will be shifted towards understanding why one would think of using a prime ideal to retain the cancellation property in A/J, i.e. to ensure that there are no divisors of zero in A/J.

Think first how the lack of divisors of zero is stated for A/J. The product of two cosets J+a and J+b in A/J is (J+a)(J+b)=J+ab. Moreover, the zero element of A/J is the coset J=J+0, so to say that A/J has no divisors of zero implies that J+ab=J\Rightarrow J+a=J or J+b=J.

Recall also that for any x\in A, x\in J \Leftrightarrow J+x=J. It then becomes obvious that the property J+ab=J\Rightarrow J+a=J or J+b=J in A/J corresponds to the property ab\in J\Rightarrow a\in J or b\in J, which is the defining property of a prime ideal J.

So it appears that in some cases it is easy to choose an ideal J whose properties are conveyed to a quotient ring A/J.


Proof: any element of a group of prime order generates the group

Let G be a group of prime order |G|=p. It will be shown that for any a\in G with a\ne e it holds that \langle a\rangle=G, where \langle a\rangle is the set generated by a and e is the identity element of G.

Since the order of group G is |G|=p<\infty, the order \mbox{ord}(a) of element a is also finite, i.e. \mbox{ord}(a)=n<\infty for some positive integer n.

Note that the set \langle a\rangle=\left\{a^0,a^1,\ldots,a^{n-1}\right\} generated by a is a subgroup of G, since \langle a\rangle is closed with respect to multiplication and with respect to inverses. Indeed, for any a^i,a^j\in\langle a\rangle, the division algorithm of integers ensures that there exist integers q,r with 0\le r <n such that a^i a^j = a^{i+j}=a^{nq+r}=(a^n)^q a^r=e a^r=a^r\in \langle a \rangle. Moreover, the inverse of any a^i\in \langle a \rangle is (a^i)^{-1}=a^{n-i}\in \langle a \rangle.

The order |\langle a \rangle| of the cyclic subgroup\langle a \rangle of G is equal to the order of generator a, that is |\langle a \rangle|=\mbox{ord}(a)=n.

By Lagrange’s theorem, n is a factor of p. Since p is a prime, n=1 or n=p. It follows from a\ne e that n\ne 1, so n=p, which means that |\langle a \rangle|=|G|.

\langle a \rangle\subseteq G and |\langle a \rangle|=|G|\Rightarrow\langle a \rangle=G.


Clarification on the proof of isomorphism of finite cyclic groups

In order to show that any two cyclic groups of finite order n are isomorphic to each other, it suffices to show that every cyclic group \langle a\rangle=\left\{a^0,a^1,a^2,\ldots,a^{n-1}\right\} of finite order n is isomorphic to the group \mathbb{Z}_n of integers modulo n with the operation of addition modulo n, where \mathbb{Z}_n=\left\{0,1,2,\ldots,n-1\right\}.

Towards this end, it can be shown that the function f:\mathbb{Z}_n\rightarrow\langle a\rangle defined as f(i)=a^i is a group isomorphism. Recalling the definition of group isomorphism, it suffices to show that f is a bijective homomorphism. It is straightforward to check that f is a bijective function. The fact that f is a homomorphism is proven also easily using laws of exponents for groups, as follows: f(i+j)=a^{i+j}=a^i a^j=f(i)f(j).

Proving the homomorphic property of f as above works well if f is taken to be f:\mathbb{Z}\rightarrow\langle a\rangle,~f(i)=a^i, that is from the additive group of integers to the cyclic group \langle a\rangle=\left\{\ldots,a^{-2},a^{-1},a^0,a^1,a^2,\ldots\right\} of infinite order generated by a. However, in the case of f:\mathbb{Z}_n\rightarrow\langle a\rangle with \langle a\rangle=\left\{a^0,a^1,a^2,\ldots,a^{n-1}\right\}, there is a slight oversight; the addition implied by f(i+j) is addition modulo n.

Let’s denote the addition modulo n by \oplus. So, now the proof of homomorphic property takes the form f(i\oplus j)=a^{i\oplus j}=a^i a^j=f(i)f(j). The identitya^{i+j}=a^i a^j is not the same asa^{i\oplus j}=a^i a^j; the former is one of the common exponent laws for groups, while the latter is not.

What needs to be shown is that the exponents of addition and of addition modulo n coincide, i.e. a^{i+j}=a^{i\oplus j}. The proof of this identity is simple, yet it is useful to be aware of such potential oversight.

According to the division algorithm for integers, there exist unique integers q and r such that i+j = nq+r and 0\le r < n. Thus, a^{i+j}=a^{nq+r}=(a^n)^q a^r. Since the cyclic group \langle a\rangle is of order n, it holds that a^n=e, where e is the identity element of \langle a\rangle. So, a^{i+j}=a^r.

Notice that r coincides with the result of the addition modulo n, that is r=i\oplus j. It follows that a^{i+j}=a^{i\oplus j}, which completes the proof.


Proof: every cycle expresses as a product of transpositions

To show that a cycle (a_1~a_2~\dots~a_{r-1}~a_r) can be written as a product of one or more transpositions, it suffices to prove the validity of the identity (a_1~a_2~\dots~a_{r-1}~a_r)=(a_r~a_{r-1})\dots (a_r~a_2)(a_r~a_1), which will be done by using mathematical induction.

Assume that the identity holds for k. It will be shown that it also holds for k+1, as in (a_1~a_2~\dots~a_{k-1}~a_k~a_{k+1})=(a_{k+1}~a_k)(a_{k+1}~a_{k-1})\dots (a_{k+1}~a_2)(a_{k+1}~a_1).

Start by noticing that (a_1~a_2~a_3~\dots~a_{k-1}~a_k~a_{k+1})=(a_2~a_3~\dots~a_{k-1}~a_k~a_{k+1})(a_{k+1}~a_1).

Due to the induction assumption, which says that a k-length cycle can be decomposed into a product of transpositions, (a_2~a_3~\dots~a_{k-1}~a_k~a_{k+1}) can be written as (a_2~a_3~\dots~a_{k-1}~a_k~a_{k+1})=(a_{k+1}~a_k)(a_{k+1}~a_{k-1})\dots (a_{k+1}~a_3)(a_{k+1}~a_2).

By combining the last two equations, the conclusion follows for k+1.


Proof: the product of disjoint cycles of a permutation is unique

Theorem 1, page 82, in the second edition of “A Book of Abstract Algebra” by Charles C. Pinter states that “Every permutation is either the identity, a single cycle, or a product of disjoint cycles”. The theorem is proved in the book. After the proof, it is stated that it is easy to see that the product of cycles is unique, except for the order of factors.

I have seen online proofs of this claim of uniqueness based on the concept of orbit. In what follows, an alternative proof is offered without relying on orbits.

Assume that there exists a permutation \pi which can be written as a product of disjoint cycles in two ways. So, there are two collections, each consisting of disjoint cycles, C=\left\{c_1,c_2,\dots,c_n\right\} and S=\left\{s_1,s_2,\dots,s_m\right\} such that C\ne S and \pi=\prod_{i=1}^{n}c_i=\prod_{i=1}^{m}s_i.

C\ne S\Rightarrow C\not\subseteq S~\mbox{or}~S\not\subseteq C. Without loss of generality, assume that C\not\subseteq S. Thus, (\exists~\mbox{cycle}~c_i \in C)(\forall~\mbox{cycles}~s_k\in S) c_i\ne s_k.

Let a_q be some element in cycle c_i. Since a_q\in c_i and the cycles in C are disjoint, a_q is permuted by \pi. Therefore, \exists~\mbox{cycle}~s_l\in S such that a_q\in s_l.

The cycles c_i and s_l are disjoint from the rest of cycles in C and S, respectively. Thus, c_i(a_q)=s_l(a_q)=\pi(a_q).

Since c_i\ne s_l, there exists an element a_p that breaks the equality arising from the successive implementation of permutation \pi in each of the two cycles c_i and s_l starting from a_q. Thereby, there exists an element a_p in c_i and in s_l with c_i(a_p)\ne s_l(a_p). This is a contradiction, since the permutation \pi is a function, so a_p should be mapped to a unique element \pi(a_p).