Let [tex]R[/tex] be a ring with [tex]1_R[/tex]. If [tex]M[/tex] is an

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In summary, a ring with 1_R is a mathematical structure that consists of a set of elements, addition and multiplication operations, and must satisfy certain commutative and distributive properties. The 1_R element, also known as the identity element, is crucial as it serves as the multiplicative identity and allows for the definition of inverses. A field is a type of ring with additional properties, such as every non-zero element having a multiplicative inverse. The M element, or module, is an object defined over a ring and allows for the generalization of vector spaces. An example of a ring with 1_R and an associated module is the set of integers with the usual addition and multiplication, and a set of polynomials with
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Let [tex]R[/tex] be a ring with [tex]1_R[/tex]. If [tex]M[/tex] is an R-module that is NOT unitary then for some [tex]m \in M[/tex], [tex]Rm = 0[/tex].

I'm pretty sure [tex]Rm = \{ r \cdot m \mid r \in R \}[/tex]. While M being not unitary means that [tex]1_R \cdot x \neq x[/tex] for some [tex]x \in M[/tex]. I'm thinking this problem should be an obvious and direct proof but I can't see it.
 
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If m ≠ 1m, then consider m - 1m ≠ 0.
 
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:blushing:

that's embarassing.
 

Related to Let [tex]R[/tex] be a ring with [tex]1_R[/tex]. If [tex]M[/tex] is an

1. What is the definition of a ring with 1_R?

A ring with 1_R is a mathematical structure that consists of a set of elements, a binary operation known as addition, and a second binary operation known as multiplication. The set of elements must form a commutative group under addition and a monoid under multiplication. Additionally, the multiplication operation must distribute over the addition operation.

2. What is the significance of the 1_R element in a ring?

The 1_R element, also known as the identity element, is a crucial component of a ring. It serves as the multiplicative identity, meaning that when any element in the ring is multiplied by 1_R, it remains unchanged. This element also allows for the definition of the inverse of an element in the ring, which is necessary for many mathematical operations.

3. What is the difference between a ring and a field?

A field is a type of ring that has additional properties. In a field, every non-zero element has a multiplicative inverse, meaning that every element can be divided by another. In contrast, not every element in a ring has a multiplicative inverse. Additionally, a field must also be commutative under multiplication, whereas a ring does not necessarily have to be.

4. What is the importance of the M element in the statement?

The M element, also known as a module, is an object that is defined over a ring. It is similar to a vector space, but instead of being defined over a field, it is defined over a ring. Modules are important in algebraic structures as they allow for the generalization of vector spaces to non-field structures.

5. Can you give an example of a ring with 1_R and an associated module?

One example of a ring with 1_R is the set of integers, [tex]\mathbb{Z}[/tex], with the usual addition and multiplication operations. An associated module over this ring could be the set of all polynomials with integer coefficients, [tex]\mathbb{Z}[x][/tex]. This module would have the same addition and multiplication operations as the ring, but with additional scalar multiplication by integers.

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