Prove that this is an ideal of a commutative ring

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In summary, we are trying to prove that the set J, defined as {rm+c | m in M and r in R}, is an ideal in the commutative ring R. To do so, we need to show that it is non-empty and that it is closed under subtraction. For the first part, we can easily see that any element x=rm+c in J since m is in M, which is an ideal in R. For the second part, we can use the fact that M is an ideal to show that rm-r'm' is in M, thus proving that J is closed under subtraction.
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annoymage
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Homework Statement



Let R be a commutative ring, [itex]c \in R[/itex], M is ideal in R

prove that [itex]J=\left\{rm+c\ |\ m \in M \ and\ r \in R \right\}[/itex] is ideal in R

Homework Equations



n/a

The Attempt at a Solution



for non-emptiness is easy

so i want to show any x=rm+c, y=r'm'+c

x-y=rm-r'm'-c+c is in J clue please T_T
 
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  • #2


x- y= r m- r' m'- c+ c= r m- r'm'. Now use the fact that m is in M which you are told is an ideal.
 
  • #3


rm-r'm' is in M right? then T_T more clue
 

Related to Prove that this is an ideal of a commutative ring

What is an ideal of a commutative ring?

An ideal of a commutative ring is a subset of the ring that is closed under addition, subtraction, and multiplication by elements of the ring. It is also absorbent, meaning that if an element of the ideal is multiplied by any element of the ring, the result is still in the ideal.

How do you prove that a subset is an ideal of a commutative ring?

To prove that a subset is an ideal of a commutative ring, you must show that it satisfies the definition of an ideal. This includes proving that it is closed under addition and multiplication, and that it is absorbent. You also need to show that it is a subset of the ring and that it contains the additive identity element of the ring.

What is the difference between a proper ideal and an improper ideal?

A proper ideal is a subset of a commutative ring that is also an ideal, but does not contain all elements of the ring. An improper ideal is a subset of a commutative ring that is also an ideal and contains all elements of the ring. In other words, a proper ideal is a strict subset of the ring, while an improper ideal is the entire ring.

Can a commutative ring have more than one ideal?

Yes, a commutative ring can have multiple ideals. In fact, every commutative ring has at least two ideals - the trivial ideal, which only contains the additive identity element, and the entire ring itself. Other commutative rings may have an infinite number of ideals.

How are ideals related to normal subgroups in group theory?

Ideals in a commutative ring are analogous to normal subgroups in group theory. Just as an ideal is a subset of a ring that is closed under addition, subtraction, and multiplication, a normal subgroup is a subset of a group that is closed under multiplication and inverse operations. Additionally, both ideals and normal subgroups are used to define quotient structures.

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