Interpret the mathematical concept of 'primary ring' as used by Zelinsky in a 1953 research paper on linearly compact rings | Step-by-Step Solution

Understanding Zelinsky's "Primary Ring" Concept

What We're Solving:

We need to understand what Zelinsky meant by "primary ring" in his 1953 paper, particularly how it relates to decomposing commutative rings into primary summands and understanding the structure of linearly compact rings.

The Approach:

To grasp this concept, we'll work from familiar ideas to Zelinsky's specialized definition. We'll start with what you might know about primary ideals, then see how Zelinsky adapted this to describe entire rings. This will help you understand both the mathematical intuition and the technical precision needed for his theorems.

Step-by-Step Solution:

Step 1: Recall Primary Ideals In commutative algebra, a primary ideal P in a ring R has the property that if xy ∈ P and x ∉ P, then some power of y is in P. Think of this as P "almost" being prime, but allowing for nilpotent elements.

Step 2: Understand Zelinsky's Ring-Level Adaptation Zelinsky took this idea and applied it to entire rings. In his context, a primary ring is a commutative ring R where the zero ideal is primary. This means:

• If xy = 0 in R and x ≠ 0, then some power y^n = 0
• In other words, R has no zero divisors except nilpotent elements

Step 3: Connect to the Radical The key insight is that in a primary ring R, every zero divisor is nilpotent, so all zero divisors lie in the nilradical (the set of all nilpotent elements). This creates a clean separation between the "nice" part of the ring and its nilpotent part.

Step 4: See the Decomposition Context Zelinsky's main result shows that certain linearly compact rings can be written as a direct sum: R = R₁ ⊕ R₂ ⊕ ... ⊕ Rₙ ⊕ S where each Rᵢ is primary and S is a radical ring (every element is nilpotent).

Step 5: Understand Why This Matters This decomposition is powerful because:

• Primary rings are "almost" integral domains (only nilpotent zero divisors)
• Radical rings are completely understood (everything is nilpotent)
• Together, they give a complete structural picture

The Answer:

A primary ring in Zelinsky's 1953 paper is a commutative ring where the zero ideal is primary - meaning every zero divisor is nilpotent. This concept allows him to decompose linearly compact rings into well-understood pieces: primary summands (which are "almost" integral domains) plus a radical component.

Memory Tip:

Think "PRIMARY = almost PRIME": A primary ring is like an integral domain, but it's allowed to have nilpotent elements that eventually disappear when raised to high enough powers. Just as primary colors are the basic building blocks for all colors, primary rings are basic building blocks in Zelinsky's decomposition theorem!

Great question about this classic paper! Understanding how mathematicians adapt familiar concepts to new contexts is a key skill in advanced algebra.