# Tag Info

1 vote

• 21.6k

### Is every vector space an injective module?

For any vector space $M$, $$\mathrm{Ext}^1_F(M,Q)=0$$ since $M$ is free, hence projective. Thus, $Q$ is injective.
• 635
Accepted

### Prime radical of a ring

It shouldn't read "Suppose $I$ is a prime ideal, then find their intersection." Rather it should read "Find all prime ideals of $R$, then find their intersection." So, you can see ...
• 154k
1 vote

### Generators and presentations of a division ring

When defining a group presentation, you work in the category of groups, so you want to force your construction to have inverses. When defining an algebra presentation, you work in the category of ...
• 26k

• 9,139

### What ring homomorphisms do induce well-defined mapping between principal ideals?

You don't specify if your rings are commutative or have unities, so let me do this as generally as possible for as long as possible... Let $R$ and $S$ be ring, and let $\phi\colon R\to S$ be a ring ...
• 402k
1 vote
Accepted

• 155
Accepted

### Ring homomorphism may not preserve $1$.

I believe a good "reason" can come from a good example. The main idea though (I think) is that just because the image of the identity needs to act like the identity (due to homomorphism ...
• 26

### Demonstrate that a set of 2x2 matrices form a finite commutative ring of unity

This seemed strange enough to me that I pulled up the book. The quoted text is exactly correct: Compare a purported ring structure on this set of four matrices to the ring $\mathbb{Z}/4\mathbb{Z}$. I ...
• 11
Accepted

### If $R$ has only a finite number of simple modules up to isomorphism, does it necessarily follow that $R$ is a semi-local ring?

There might be much simpler examples, but I found the following example on the DaRT Database of Ring Theory. It's a bit complicated to define from scratch (it depends on Kolchin's definition of a ...
• 29.4k
1 vote
Accepted

### Euclidean algorithm in commutative rings with unity

Your intuition seems right, but it may be easier to use what we have in $\mathbb{Z}$ rather than try to copy-paste it onto a new ring. This will avoid concerns about new factors (e.g. presence of ...
• 2,444

### Proving a module is injective (or not) and failed attempt using Baer's Criterion.

Again I prefer to use $F_2$ for clarity and ease of typing. One thing that simplifies the situation here is how many summands of $R$ there are. Each one that is a direct summand of $_RR$ easily ...
• 154k
1 vote

### Proving a module is injective (or not) and failed attempt using Baer's Criterion.

$\require{AMScd}$ This problem can be easily solved by using the language of quiver representations for finite-dimensional algebras over a field. If you are not familiar with this theory, it is a very ...
1 vote

### Is every sufficiently large gaussian integer the sum of 3 cubes ? $a + b i = (c + di)^3 + (e + fi)^3 + (g + hi)^3$?

I can answer the second question . . . Let $\omega$ be a primitive cube root of unity. Claim:$\;$For integers $a,b$ with $a$ odd and $b\equiv 2\;(\text{mod}\;4)$, there are no integers $c,d,e,f$ for ...
• 59k

• 402k
1 vote
Accepted

### If $f: A \to B$ is a homomorphism, then a prime $\mathfrak{p} \subseteq A$ is the contraction of a prime in $B$ iff $\mathfrak{p}^{ec} = \mathfrak{p}$

Assume that $\mathfrak{p}^{ec} = \mathfrak{p}$. We need to show that there is a prime that contracts to $\mathfrak{p}$. As the OP pointed out, $\mathfrak{p}^e$ may not be prime in $B$. We will find a ...
• 98

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