Err is Human

Err is Human

M. Velrajan 

When we learn from a book or any source

we should not take it for granted that what

is given there is correct.

We have to apply our mind, knowledge.

Suppose we learn about  Euclidean rings

and the following

Unique Factorization Theorem from

Topics in Algebra,

by I. N. Herstein, second edition.

We have to understand the statement

and each and every step of the proof

and we have to prepare notes.

After understanding the statement,

suppose we write the proof as follows : 

 

Proof Suppose that

a =  𝜋1𝜋2…. 𝜋n = 𝜋′1𝜋′2…. 𝜋′m. 

Since 𝜋1 | 𝜋1𝜋2…. 𝜋n, 𝜋1 | 𝜋′1𝜋′2….𝜋′m.

Since 𝜋1 is prime, 𝜋1  | 𝜋′i for some

1 ≤ i ≤ m.

Since 𝜋1 and 𝜋′i are

both prime elements

of R and 𝜋1 | 𝜋′i, 𝜋1 and 𝜋′i are

associates

and 𝜋′i = u1𝜋1, where u1 is a unit in R.

Hence 

𝜋1𝜋2…𝜋n = u1𝜋1𝜋′1𝜋′2 . . .𝜋′i-1𝜋′i+1 . . . 𝜋′m

= 𝜋1u1𝜋′1𝜋′2 . . .𝜋′i-1𝜋′i+1 . . . 𝜋′m. 

By the left cancellation in R,  

𝜋2…𝜋n = u1𝜋′2 . . .𝜋′i-1𝜋′i+1 . . . 𝜋′m.

Hence

u1-1𝜋2….𝜋n = 𝜋′2 . . .𝜋′i-1𝜋′i+1 . . . 𝜋′m.

And hence we can proceed as  above

with

𝜋2  | 𝜋′2 . . .𝜋′i-1𝜋′i+1 . . . 𝜋′m. 

If n < m then proceeding thus, we get

each 𝜋i is an associate of some 𝜋′j  

and after n steps we get

1 = u1u2…un𝜋′j1…𝜋′j(m-n).

Hence 𝜋′j(m-n) is a unit, 

which is a contradiction,

since 𝜋′j(m-n) is a

prime element.

Hence n ≥ m and each each 𝜋i is an

associate of some 𝜋′j, 1 ≤ j ≤ m. 

Now by  considering

𝜋′1 |𝜋1𝜋2… 𝜋m and

proceeding as above, we get m ≥ n and

each 𝜋′i | 𝜋j for some j.

Therefore, n = m and each each 𝜋′i is

an associate of some 𝜋j, 1 ≤ j ≤ n.   

Oh, something is wrong. 

Only if  n < m after n steps we get

1 = u1u2…un𝜋′j1…𝜋′j(m-n)

and get a contradiction.

Since we get a contradiction, n ≥ m.

But in the book we have : 

“After n steps . . . (excess of m over n).” is

correct. But in

“This would force n ≤ m since 𝜋′  are not units”,

n ≤ m is not correct.

It should have been n ≥ m.

Similarly, m ≥ n, . . .  . 

This might be a typographic error.

So, dear young learners of Mathematics,

don't take it for granted wherever you

get it. Apply mathematical thinking and

check for the validity. 

So, don't take all I have shared with you

as it is, you yourself justify.

Err is Human.