Why call Integer Z a Ring ?

​Does below picture looks like a ring, or a clock if it is Z12 = {1 2 3 … 12 = 0}.

Integers (Z) have 3 operations : {+ – x} but not {÷} (or multiplicative inverse) – otherwise 2 integers divide would give a fraction (Q) which falls out of Z family.

 
An important property of “special ring” Zp for any prime number p, eg. Z2 = {0 , 1} has additional “÷” operation (or multiplicative inverse) besides {+ – ×}, so it is a “Field”.

Example: Z5 = {0 1 2 3 4}
2.3 = 6 = 1 (mod 5)
=> 2 has a multiplicative inverse 3 in Z5, vice versa.

Zp is useful in encryption coding “RSA” using prime number theorems such as The Chinese Reminder Theorem and Fermat’s Little Theorem.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s