Exhibit 32

THE LUBRIZOL CORPORATION

Certification of Chief Executive Officer and Chief Financial Officer of

The Lubrizol Corporation Pursuant to 18 U.S.C. Section 1350

I certify that, to the best of my knowledge and belief, the Quarterly Report on Form 10-Q of The Lubrizol Corporation for the period ending March 31, 2004:

(1) fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934; and

(2) the information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of The Lubrizol Corporation.

/s/    James L. Hambrick

James L. Hambrick

Chief Executive Officer and President

May 6, 2004

I certify that, to the best of my knowledge and belief, the Quarterly Report on Form 10-Q of The Lubrizol Corporation for the period ending March 31, 2004:

(3) fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934; and

(4) the information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of The Lubrizol Corporation.

/s/    Charles P. Cooley

Charles P. Cooley

Chief Financial Officer

May 6, 2004

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