Intereting Posts

$a,b,c,d\ne 0$ are roots (of $x$) to the equation $ x^4 + ax^3 + bx^2 + cx + d = 0 $
What is the spectral theorem for compact self-adjoint operators on a Hilbert space actually for?
About polar coordinates in high dimensions
$-4\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5)=S$
How to find the maximum value for the given $xcos{\lambda}+ysin{\lambda}$?
Complex part of a contour integration not using contour integration
Recovering a finite group's structure from the order of its elements.
The intersection of two Sylow p-subgroups has the same order
Proof of Inequality using AM-GM
Projective Nullstellensatz
If I have the presentation of a group, how can I find the commutator subgroup of it?
Combinatorial argument for $1+\sum_{r=1}^{r=n} r\cdot r! = (n+1)!$
Let $f$ be a continuous function satisfying $\lim \limits_{n \to \infty}f(x+n) = \infty$ for all $x$. Does $f$ satisfy $f(x) \to \infty$?
Sequence of Rationals Converging to a Limit
Two homogenous system are equivalent if they have the same answer

$$xy\in\mathfrak q\:\Rightarrow\:\text{either $x\in\mathfrak q$ or $y^n\in\mathfrak q$ for some $n\gt0$}.$$

Primary ideals can be regard as the generalization of prime ideals and radical.

But why it’s defined like that? It’s not symmetry.

Why not define like that:

- Question about fields and quotients of polynomial rings
- Can we contruct a basis in a finitely generated module
- Problem related polynomial ring over finite field of intergers
- Definition of a Group in Abstract Algebra Texts
- Any subgroup of index $p$ in a $p$-group is normal.
- Smallest example of a group that is not isomorphic to a cyclic group, a direct product of cyclic groups or a semi direct product of cyclic groups.

$$xy\in\mathfrak q\:\Rightarrow\:\text{either $x^n\in\mathfrak q$ or $y^n\in\mathfrak q$ for some $n\gt0$}.$$

- $M_a =\{ f\in C |\ f(a)=0 \}$ for $a$ $\in$ $$. Is $M_a$ finitely generated in $C$?
- Does every Abelian group admit a ring structure?
- Efficiently find the generators of a cyclic group
- The fix points of the Möbius transformations are the eigenspace of a certain matrix.
- Show $\vert G \vert = \vert HK \vert$ given that $H \trianglelefteq G$, $G$ finite and $K \leq G$.
- Odd/Even Permutations
- Center of Weyl algebra over a field of characterstic $0$?
- Does $R \cong S$ imply $R \cong S$?
- A Book for abstract Algebra
- Commutativity of “extension” and “taking the radical” of ideals

I’ve seen this question many times. The problem is that the second definition is strictly weaker than the first.

Consider $(x^2,xy)$ in the ring $F[x,y]$ where $F$ is a field. According to the normal definition, it is not primary since it doesn’t contain any powers of $y$ and doesn’t contain $x$.

However, it does satisfy the second definition. If $ab$ is in $(x^2, xy)$, then $x$ divides one of $a$ or $b$, and then that element’s square is in this ideal.

The ordinary definition links zero divisors to nilpotent elements. In the quotient of a ring by a primary ideal, the elements are divided into regular elements and nilpotent elements. Said another way, zero divisors are nilpotent in such a ring.

Another way to think about primary ideals is that the condition is one half of primeness. An ideal is prime iff it is radical and primary. (But actually, you could replace primary with your definition and that would still hold!)

- What is the equation for a 3D line?
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- What are the “real math” connections between Euclidean Geometry and Complex Numbers?
- What are the postulates that can be used to derive geometry?
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- Discussion on even and odd perfect numbers.
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- Group with order $p^2$ must be abelian . How to prove that?
- Precision of operations on approximations