19.2.3. Examples of Rings

So, let us see some examples for ring. So, recall our set ℤ is the set of integers 0 to N - 1 and N suppose I take 2 operations here: plus operation is the addition modulo N (+) and my multiplication operation is multiplication modulo N (∙) and my claim is that with respect to these 2 operations, my set ℤ satisfies all the ring axioms.

In this chunk, we introduce the set of integers, denoted as ℤ, defined as {0, 1, 2, ..., N-1}. Two specific operations are defined for this set: addition and multiplication, both performed modulo N. Modulo operation means when the result exceeds N, we wrap around. The statement claims that ℤ with these operations meets all criteria for being classified as a ring.

Imagine you're playing a game with a circular board that has N positions. Whenever you move, you count the steps but start over at position 0 after reaching N. That’s like our addition modulo N. If you have 3 positions and move from position 2 to position 3, you would end up at position 0 instead.

So, it is easy to verify that indeed the collection 0 to N - 1 along with the operation addition modulo N constitutes an abelian group we had already proved in our earlier discussion. And it is also easy to verify that if we consider the multiplication modulo N operation then it satisfies the closure property, the operation is associative and identity element is actually the numeric 1, integer 1 which is actually present in your set ℤ.

This chunk is focused on verifying that the set ℤ with operations modulo N indeed forms a ring. The addition operation follows the rules of an abelian group, such as closure (the result is still in the set) and an identity element (0 in this case for addition). The multiplication also satisfies necessary properties like closure, associativity, and identity with 1, validating these operations as consistent under modulo N.

Think of a group of friends passing a ball in a circle. When one friend passes to another, they always end up with someone in the group, just like adding numbers modulo N always results in another number in the set. The identity is like having a designated spot where no one needs to make a pass; that spot is where everyone starts counting.

Because typically in our computers in our programming we have registers we have either 32 bit registers, 64 bit registers or even if you have powerful processor then you have 128 bit registers, where you can save values using either 32 bit or 64 bits or 128 bits and so on and if you add any two 32 bit number, then again you get back a 32 bit answer.

This part discusses the practical application of rings in computer science. In computers, there are specific bit sizes for registers (e.g., 32-bit), and any arithmetic performed exceeds this size will wrap around using modular arithmetic. This ensures that the outcomes remain within a valid range that the computer can handle according to the defined operations, ensuring consistent behavior across calculations.

Consider a digital scoreboard that can only display 0 to 9. If you score 7 points and then add 5 more, it would show 2 instead of 12. It wraps around, similar to how computers handle numbers with constrained bit sizes, employing modular arithmetic to maintain manageable results.

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