Ideals and Residue class ring

• EV33
In summary, the conversation discusses the conditions for a commutative ring R to have ideals I, J, and M, with M being contained in J if M/I is a subset of J/I. The conversation then presents a proof by contradiction to show that M must be contained in J. However, there is a potential flaw in the reasoning, as the assumption that m+I and j+I are different sets may not be necessarily true.

Homework Statement

I am curious,
if I,J, and M are ideals of the commutative ring R, and M/I$\subseteq$J/I, then M$\subseteq$J

Homework Equations

M/I = { m+I : m is in M}

J/I = { j+I : j is in J}

I$\subseteq$R is an ideal if
1.) if a and b are in I then a+b is in I
2.) if r is in R and a is in I then a*r is in I

The Attempt at a Solution

Assume M/I$\subseteq$J/I,
and M is not contained in J.

Since M is not contained in J then there exists a m in M that is not in J.
Then m+I is in M/I but not in J/I.

This is a contradiction to the hypothesis that M/I$\subseteq$J/I. Thus M is contained in J. (QED)

Any flaws?

Last edited:
EV33 said:
Since M is not contained in J then there exists a m in M that is not in J.
Then m+I is in M/I but not in J/I.

This step isn't necessarily obvious. The coset m+I can go by many different names. How do you know it isn't the same set as j+I for some j in J?

I suggest assuming that m+I = j+I for some j in J, and deriving a contradiction.

1. What is an ideal in a residue class ring?

An ideal in a residue class ring is a subset of the ring that is closed under addition and multiplication by elements in the ring. It is also a subgroup of the additive group of the ring. Ideals are important in residue class rings because they help to identify common properties and structures within the ring.

2. What is a residue class ring?

A residue class ring, also known as a quotient ring, is a mathematical structure that extends the concept of modular arithmetic. It is created by taking a ring and defining an equivalence relation on its elements, which results in a new ring with elements that represent the equivalence classes.

3. What is the significance of residue class rings in abstract algebra?

Residue class rings are important in abstract algebra because they allow for the study of abstract algebraic structures without relying on specific elements or operations. They also provide a framework for understanding the structure of rings and their properties, such as ideals and homomorphisms.

4. How are ideals and residue class rings related?

Ideals and residue class rings are closely related because ideals are used to construct residue class rings. In fact, residue class rings are often defined as the set of all possible cosets of an ideal in a ring. Additionally, ideals play a crucial role in the study of residue class rings, as they help to identify important structures and properties within the ring.

5. What are some practical applications of residue class rings?

Residue class rings have many practical applications, particularly in fields such as cryptography and coding theory. In cryptography, residue class rings are used to create public key systems, such as the RSA algorithm. In coding theory, residue class rings are used to construct error-correcting codes that are used in telecommunications and data storage.

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