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The Jacobson radical is the intersection of all maximal left ideals, equivalently the intersection of the annihilators of all simple left modules.

Morita-equivalent rings have equivalent module categories. In particular, $R$ and ${\mathrm{Mat}}_n(R)$ are Morita equivalent.

A left primitive ring has a faithful simple left module.

The first Weyl algebra over a field $k$ isIn characteristic zero, it acts faithfully on $k[x]$: $x$ multiplies by the variable and $\partial$ differentiates. The defining relation is the product rule, since $\partial (xf)-x\partial (f)=f$. Every element has a unique ordered expression ${\sum }_{i,j}{a}_{ij}{x}^i{\partial }^j$, giving the ordered monomial basis of a Weyl algebra.

The higher Weyl algebra $A_n(k)$ has commuting coordinates $x_i$, commuting derivatives ${\partial }_i$, and $[{\partial }_i,x_j]={\delta }_{ij}$. It organizes polynomial-coefficient differential operators. The abstract algebra remains meaningful in positive characteristic, but its usual action on polynomials is no longer faithful: ${\partial }^p$ acts as zero in characteristic $p$ while remaining a nonzero ordered monomial. See Weyl algebra in positive characteristic for the resulting finite-dimensional representations.

A ring is right Noetherian when every ascending chain of right ideals stabilizes.

A ring is left Noetherian when every ascending chain of left ideals stabilizes, equivalently when every left ideal is finitely generated.

Suspension rheology relates particle concentration, particle pressure, shear stress, deformation rate, and fluid migration in a mixture of particles and a liquid.

Viscoelasticity combines recoverable elastic response with dissipative and time-dependent response. Two specimens may react similarly to a rapid pull but behave differently when the same load is maintained. In a material creep test, constant stress produces evolving strain; in a stress relaxation test, fixed strain produces evolving stress.

The Maxwell model makes the distinction explicit by placing an elastic spring and viscous element in series. In simple shear strain it obeysAfter a small step to fixed ${\gamma }_0$, its shear stress decays as $\tau (t)=G{\gamma }_0{e}^{-t/\lambda }$ with viscoelastic relaxation time $\lambda =\eta /G$. Under constant shear stress ${\tau }_0$, it instead continues to deform, with $\gamma (t)={\tau }_0/G+{\tau }_{0}t/\eta$. These are predictions of this particular model: a Kelvin-Voigt model has bounded creep and different short-time behavior. Small-amplitude oscillatory shear measures the same competition through storage modulus and loss modulus, rather than assuming that one constant dynamic viscosity characterizes the material.

A power-law fluid has shear stress $\tau =K|\dot{\gamma }{|}^{n-1}\dot{\gamma }$. It is shear thinning for $n<1$, Newtonian for $n=1$, and shear thickening for $n>1$.

A yield-stress fluid can sustain a finite shear stress without steady flow, but flows when the applied loading exceeds its yield stress. The ideal Bingham plastic treats the unyielded region as rigid and gives, after yield,Here $\dot{\gamma }$ is the shear rate, ${\tau }_y$ the yield stress and ${\eta }_p$ the post-yield dynamic viscosity. Real materials can undergo small elastic deformations below the flow threshold; an ideal rigid branch does not describe every part of their response.

A pressure-driven experiment makes the spatial distinction visible. Where the local shear stress is below threshold, a plug flow of a yield-stress fluid can move without internal shearing, while adjacent yielded layers deform. If every point remains below threshold and the walls have no wall slip, the ideal material does not flow. Slowly evolving microstructure and thixotropy make the apparent threshold depend on preparation, waiting time and the loading protocol.

A local field is a nondiscrete locally compact topological field. A non-Archimedean local field is a finite extension of ${\mathbb{Q}}_p$ or of ${\mathbb{F}}_p((t))$.

Fix a prime number $p$. Write a nonzero rational number as $x=p^{k}a/b$, where $k\in \mathbb{Z}$ and the nonzero integers $a,b$ are not divisible by $p$. The exponent $k$ is uniquely determined. Its p-adic valuation is this signed exponent $v_p(x)=k$: equivalently, for any nonzero integer numerator $A$ and denominator $B$,Factors of $p$ in the numerator decrease the p-adic absolute value; factors in the denominator increase it. For $p=2$ the computations are:

This is an absolute value on a field measuring divisibility, rather than the magnitude on a real number number line. It satisfiesThe proof comes directly from p-adic valuations: exponents add in a product of numbers, and extracting a common power of $p$ from a sum gives $v_p(x+y)\ge \min \{v_p(x),v_p(y)\}$. If the two p-adic valuations differ, the remaining term with smaller exponent cannot cancel modulo $p$, so equality holds. Thus the p-adic absolute value is a Non-Archimedean absolute value. ^[ref]

The metric $d_p(x,y)=|x-y{|}_p$ makes two rational numbers close when their difference contains a high power of $p$. This packages successive modular congruences into a numerical error estimate, allowing convergence in a metric space and convergent series to encode exact arithmetic information.

For an ordered abelian group $\Gamma$, the Hahn series field $k((t^{\Gamma }))$ consists of formal sums with well-ordered support in $\Gamma$. The least exponent defines a valuation with value group $\Gamma$.

An Artinian ring satisfies the descending chain condition on ideals. A commutative ring is Artinian exactly when it is Noetherian and has Krull dimension zero.

The subring $R=K+XYK[X,Y]$ is not Noetherian. Its idealsform a strict ascending chain because multiplying by a nonconstant element of $R$ introduces at least a second power of $X$.

For a nonsquare integer $d$,The integers are Noetherian, the Hilbert basis theorem makes $\mathbb{Z}[X]$ Noetherian, and quotients preserve Noetherianity.

A polynomial ring over a Noetherian ring is Noetherian.

If a ring $R$ is finitely generated as an abelian group, then it is Noetherian. Every ideal is an additive subgroup of that finitely generated abelian group and is therefore generated by finitely many elements as an abelian group. The same elements generate it as an $R$-ideal because integer multiples are already scalar multiples by elements of $R$.

The ascending chain condition requires every chainof ideals to stabilize. A ring is Noetherian exactly when it satisfies this condition, equivalently when every ideal is finitely generated.

A Noetherian local ring is a Noetherian ring with a unique maximal ideal.